MOBILE BEING IN GENERAL
Need for defining motion and things related to it.
Chapter 1 Ἐπεὶ δ' ἡ φύσις μέν ἐστιν ἀρχὴ κινήσεως καὶ μετα βολῆς, ἡ δὲ μέθοδος ἡμῖν περὶ φύσεώς ἐστι, δεῖ μὴ λανθάνειν τί ἐστι κίνησις· ἀναγκαῖον γὰρ ἀγνοουμένης αὐτῆς ἀγνοεῖσθαι καὶ τὴν φύσιν. Nature has been defined as a 'principle of motion and change', and it is the subject of our inquiry. We must therefore see that we understand the meaning of 'motion'; for if it were unknown, the meaning of 'nature' too would be unknown. διορισαμένοις δὲ περὶ κινήσεως πειρατέον τὸν αὐτὸν ἐπελθεῖν τρόπον περὶ τῶν ἐφεξῆς. δοκεῖ δ' ἡ κίνησις εἶναι τῶν συνεχῶν, τὸ δ' ἄπειρον ἐμφαίνεται πρῶτον ἐν τῷ συνεχεῖ· διὸ καὶ τοῖς ὁριζομένοις τὸ συνεχὲς συμβαίνει προσχρήσασθαι πολλάκις τῷ λόγῳ τῷ τοῦ ἀπείρου, ὡς τὸ εἰς ἄπειρον διαιρετὸν συνεχὲς ὄν. πρὸς δὲ τούτοις ἄνευ τόπου καὶ κενοῦ καὶ χρόνου κίνησιν ἀδύνατον εἶναι. δῆλον οὖν ὡς διά τε ταῦτα, καὶ διὰ τὸ πάντων εἶναι κοινὰ καὶ καθόλου ταῦτα, σκεπτέον προχειρισαμένοις περὶ ἑκάστου τούτων (ὑστέρα γὰρ ἡ περὶ τῶν ἰδίων θεωρία τῆς περὶ τῶν κοινῶν ἐστιν)· καὶ πρῶτον, καθάπερ εἴπαμεν, περὶ κινήσεως. When we have determined the nature of motion, our next task will be to attack in the same way the terms which are involved in it. Now motion is supposed to belong to the class of things which are continuous; and the infinite presents itself first in the continuous—that is how it comes about that 'infinite' is often used in definitions of the continuous ('what is infinitely divisible is continuous'). Besides these, place, void, and time are thought to be necessary conditions of motion. Clearly, then, for these reasons and also because the attributes mentioned are common to, and coextensive with, all the objects of our science, we must first take each of them in hand and discuss it. For the investigation of special attributes comes after that of the common attributes. To begin then, as we said, with motion. ἔστι δὴ [τι] τὸ μὲν ἐντελεχείᾳ μόνον, τὸ δὲ δυνάμει καὶ ἐντελεχείᾳ, τὸ μὲν τόδε τι, τὸ δὲ τοσόνδε, τὸ δὲ τοιόνδε, καὶ τῶν ἄλλων τῶν τοῦ ὄντος κατηγοριῶν ὁμοίως. τοῦ δὲ πρός τι τὸ μὲν καθ' ὑπεροχὴν λέγεται καὶ κατ' ἔλλειψιν, τὸ δὲ κατὰ τὸ ποιητικὸν καὶ παθητικόν, καὶ ὅλως κινητικόν τε καὶ κινητόν· τὸ γὰρ κινητικὸν κινητικὸν τοῦ κινητοῦ καὶ τὸ κινητὸν κινητὸν ὑπὸ τοῦ κινητικοῦ. We may start by distinguishing (1) what exists in a state of fulfilment only, (2) what exists as potential, (3) what exists as potential and also in fulfilment—one being a 'this', another 'so much', a third 'such', and similarly in each of the other modes of the predication of being. Further, the word 'relative' is used with reference to (1) excess and defect, (2) agent and patient and generally what can move and what can be moved. For 'what can cause movement' is relative to 'what can be moved', and vice versa. οὐκ ἔστι δὲ κίνησις παρὰ τὰ πράγματα· μεταβάλλει γὰρ ἀεὶ τὸ μεταβάλλον ἢ κατ' οὐσίαν ἢ κατὰ ποσὸν ἢ κατὰ ποιὸν ἢ κατὰ τόπον, κοινὸν δ' ἐπὶ τούτων οὐδὲν ἔστι λαβεῖν, ὡς φαμέν, ὃ οὔτε τόδε οὔτε πο (201a.) σὸν οὔτε ποιὸν οὔτε τῶν ἄλλων κατηγορημάτων οὐθέν· ὥστ' οὐδὲ κίνησις οὐδὲ μεταβολὴ οὐθενὸς ἔσται παρὰ τὰ εἰρημένα, μηθενός γε ὄντος παρὰ τὰ εἰρημένα. Again, there is no such thing as motion over and above the things. It is always with respect to substance or to quantity or to quality or to place that what changes changes. But it is impossible, as we assert, to find anything common to these which is neither 'this' nor quantum nor quale nor any of the other predicates. Hence neither will motion and change have reference to something over and above the things mentioned, for there is nothing over and above them. ἕκαστον δὲ διχῶς ὑπάρχει πᾶσιν, οἷον τὸ τόδε (τὸ μὲν γὰρ μορφὴ αὐτοῦ, τὸ δὲ στέρησις), καὶ κατὰ τὸ ποιόν (τὸ μὲν γὰρ λευκὸν τὸ δὲ μέλαν), καὶ κατὰ τὸ ποσὸν τὸ μὲν τέλειον τὸ δ' ἀτελές. ὁμοίως δὲ καὶ κατὰ τὴν φορὰν τὸ μὲν ἄνω τὸ δὲ κάτω, ἢ τὸ μὲν κοῦφον τὸ δὲ βαρύ. ὥστε κινήσεως καὶ μεταβολῆς ἔστιν εἴδη τοσαῦτα ὅσα τοῦ ὄντος. Now each of these belongs to all its subjects in either of two ways: namely (1) substance—the one is positive form, the other privation; (2) in quality, white and black; (3) in quantity, complete and incomplete; (4) in respect of locomotion, upwards and downwards or light and heavy. Hence there are as many types of motion or change as there are meanings of the word 'is'.
Postquam philosophus determinavit de principiis rerum naturalium, et de principiis huius scientiae, hic incipit prosequi suam intentionem determinando de subiecto huius scientiae, quod est ens mobile simpliciter. 275. Having settled the question of the principles of natural things (Book I), and that of the principles of this science (Book II), the Philosopher here begins to pursue his original plan, which is to arrive at conclusions concerning the subject of this science, mobile being taken absolutely. Dividitur ergo in partes duas: in prima determinat de motu secundum se; in secunda de motu per comparationem ad moventia et mobilia, ibi: omne quod movetur etc., libro VII. The treatment, then, is divided into two parts: In the first he concludes with respect to motion in itself (Books III-VI); In the second he concludes with respect to motion in relation to movers [things moving others] and things movable [things which others move] (Book VII). Prima dividitur in duas: in prima determinat de ipso motu; in secunda de partibus eius, in quinto libro, ibi: transmutatur autem et cetera. The first part is divided into two: He concludes in regard to motion itself (Books III-IV); He concludes in regard to its parts (Book V). Circa primum duo facit: primo dicit de quo est intentio; secundo exequitur, ibi: est quidem igitur et cetera. As to the first, he does two things: He states what is under investigation; He follows it out, at 279. Circa primum duo facit: primo dicit de quo principaliter intendit; secundo ponit quaedam ei adiuncta, quae ex consequenti intenduntur, ibi: determinantibus autem et cetera. With reference to the first of these, he does two things: He states that concerning which he intends to treat principally; He sets down certain things which adjoin thereto, with which he will be subsequently concerned, at 277. Circa primum utitur tali ratione natura est principium motus et mutationis, ut ex definitione in secundo posita patet (quomodo autem differant motus et mutatio, in quinto ostendetur): et sic patet quod ignorato motu, ignoratur natura, cum in eius definitione ponatur. Cum ergo nos intendamus tradere scientiam de natura, necesse est notificare motum. 276. As to the first  he uses the following argument: Nature is the principle of motion and change, as is evident from the definition set down in Book II. (But how motion and change differ, will be shown in Book V.) And thus it is evident that if one does not know motion, one does not know nature, since the former [motion] is placed in the definition of the latter [nature]. Since, therefore, we intend to present the science of nature, we must make motion understood. Deinde cum dicit: determinantibus autem etc., adiungit quaedam quae concomitantur motum: et utitur duabus rationibus, quarum prima talis est.. 277. Then  he adds certain things which accompany motion. And he employs two sets of reasons [for including them], the first of which is as follows [the second at no. 2778, below]: Quicumque determinat de aliquo, oportet quod determinet ea quae consequuntur ipsum: subiectum enim et accidentia in una scientia considerantur Whoever determines something, must determine those things which follow upon it—for the subject and its accidents [Properties] are considered in a single science. Sed motum consequitur infinitum intranee, quod sic patet. But the infinite follows upon motion intrinsically, as the following makes plain: Motus enim est de numero continuorum, quod infra patebit in sexto: infinitum autem cadit in definitione continui. Motion is of the number of continuous things, as will be evident below in Book VI (l.6). But “infinite” enters into the definition of “ continuum. ” Et addit primo, quia infinitum quod est in additione numeri, causatur ex infinito quod est in divisione continui. Et quod infinitum cadat in definitione continui, ostendit quia multoties definientes continuum utuntur infinito; utpote cum dicunt quod continuum est quod est divisibile in infinitum. And he [Aristotle] adds “first of all,” because the infinite which is found in the addition of number, is caused from the infinite which is in the division of the continuum. And that the infinite enters [first of all] into the definition of the continuum, he shows from the fact that those defining the continuum often use “infinite”—as, for example, when they say that the “continuum” is that which is “divisible to infinity.” Et dicit multoties, quia invenitur etiam alia definitio continui, quae ponitur in praedicamentis: continuum est cuius partes ad unum terminum communem copulantur. And he [Aristotle] says “often,” since there is also found another definition of the continuum, which is given in the Predicaments [or Categories ]: the “continuum” is that “whose parts are joined at a common boundary.” Differunt autem hae duae definitiones. Continuum enim, cum sit quoddam totum, per partes suas definiri habet: partes autem dupliciter comparantur ad totum, scilicet secundum compositionem, prout ex partibus totum componitur; et secundum resolutionem, prout totum dividitur in partes. Now these two definitions differ. For the continuum, since it is a certain whole, is properly defined through its parts. But parts are compared to the whole in a twofold way, namely, as its components, i.e., according to composition, insofar as the whole is composed out of the parts; and as its resolutes, i.e., according to resolution, insofar as the whole is divided into the parts. Haec igitur definitio continui data est secundum viam resolutionis; quae autem ponitur in praedicamentis, secundum viam compositionis. The present definition, therefore, of the continuum, is given according to the mode of resolution [division into parts]; while that which is set down in the Predicaments is according to the mode of composition [composition out of parts]. Sic igitur patet quod infinitum consequitur motum intranee. Hence it is clear that the infinite follows upon motion intrinsically. Quaedam autem consequuntur motum extrinsece, sicut exteriores quaedam mensurae, ut locus et vacuum et tempus. But there are some things which follow upon motion extrinsically, as certain external measures: such as place, and the void, and time. Nam tempus est mensura ipsius motus: mobilis vero mensura est locus quidem secundum veritatem, vacuum autem secundum opinionem quorundam: et ideo subiungit quod motus non potest esse sine loco, vacuo et tempore. For time is the measure of motion itself; while the measure of the mobile thing is indeed place according to truth, but the void according to the opinion of some. And therefore he adds that motion cannot be without place, the void, and time. Nec impedit quod non omnis motus est localis; quia nihil movetur nisi in loco existens: omne enim corpus sensibile est in loco, et huius solius est moveri. Nor does the fact that not all motion is local affect this; since nothing is moved which is not in place. For every sensible body is in place, and to it [sensible body] alone does it belong to be moved. Motus etiam localis est primus motuum, quo remoto removentur alii, ut infra patebit in octavo. Likewise, local motion is the first of motions, which, when it is removed, the other motions are removed, as will be evident below in Book VIII (l.14). Sic igitur patet quod praedicta quatuor consequuntur motum, unde pertinent ad considerationem philosophi naturalis propter rationem praedictam. It is thus clear that the four above-mentioned properties are consequent upon motion; whence they pertain to the consideration of the natural philosopher for the aforesaid reason. Et etiam propter aliam quam consequenter subiungit, quia praedicta sunt communia omnibus rebus naturalibus. 278. This is also true for yet another reason which he [Aristotle] adds subsequently: namely, because the aforesaid are common to all natural things. Unde cum determinandum sit in scientia naturali de omnibus rebus naturalibus, praedeterminandum est de quolibet istorum: quia speculatio quae est de propriis, est posterior ea quae est de communibus, ut in principio dictum est. Sed inter haec communia prius determinandum est de motu; quia alia consequuntur ad ipsum, ut dictum est. Accordingly, since it is the task of natural science to reach conclusions concerning all natural things, one must therefore first determine concerning each of these [four]. For the speculation which is directed toward proper things, comes after that which is of common things, as was stated in the beginning [Book I, l.1, no. 6]. But among all these common things, one must first reach conclusions concerning motion itself, because the other things follow upon it as was stated [in the preceding no]. Deinde cum dicit: est quidem igitur aliquid etc., exequitur propositum. 279. Then  he puts his plan into execution: Et primo determinat de motu et infinito, quod intranee motum consequitur; secundo de aliis tribus, quae consequuntur ipsum extrinsece, in quarto libro, ibi: similiter autem necesse et cetera. He reaches conclusions concerning motion, and the infinite, which follows motion intrinsically; He does the same for the other three, which follow motion extrinsically, and this he does in Book IV. Prima dividitur in duas: in prima determinat de motu; in secunda de infinito, ibi: quoniam autem de natura et cetera. The first treatment is divided into two parts: He concludes with respect to motion; He does the same for the infinite, at 326. Circa primum duo facit: primo praemittit quaedam ad investigandum definitionem motus; secundo definit motum, ibi: diviso autem secundum unumquodque et cetera. With respect to the first of these, he does two things: He prefaces his treatment with certain considerations requisite for investigating the definition of motion; He defines motion, at 283. Circa primum duo facit: primo enim praemittit quasdam divisiones: quia via ad inveniendum definitiones convenientissima est per divisiones, ut patet per philosophum in II Poster. et in VII Metaphys.; secundo ostendit quod motus in praedictas divisiones cadit, ibi: non est autem motus et cetera. As to the first of these, he does two things: He sets down in advance certain divisions, since the most suitable path towards finding definitions is through division, as is clear from the Philosopher in Posterior Analytics II (l.14 ff.), and in Metaphysics VII (l.12); He shows that motion falls within the aforesaid divisions, at 281. Circa primum ponit tres divisiones: quarum prima est quod ens dividitur per potentiam et actum. Et haec quidem divisio non distinguit genera entium: nam potentia et actus inveniuntur in quolibet genere. Secunda divisio est prout ens dividitur secundum decem genera: quorum unum est hoc aliquid, idest substantia, aliud quantum vel quale, aut aliquod aliorum praedicamentorum. Tertia divisio est unius generis entium, scilicet eius quod est ad aliquid. Nam motus aliquo modo ad hoc genus pertinere videtur, inquantum movens refertur ad mobile. 280. With respect to the first of these, he sets down three divisions:
- The first of these is that being is divided by potency and act. Now this division does not distinguish beings into genera—for potency and act are found in every genus.
- The second division is of being as divided according to the ten genera: the first of these is “this something,” i.e., substance; others are: how much [i.e., quantity], or how [quality], or some other of the Predicaments.
- The third division is of one genus of beings, namely, of the one which is “to something” [relation]. For motion seems in a certain way to pertain to this genus, insofar as the mover is referred to the movable thing.
Ad huius igitur tertiae divisionis intellectum, considerandum est quod, cum relatio habeat debilissimum esse, quia consistit tantum in hoc quod est ad aliud se habere, oportet quod super aliquod aliud accidens fundetur; quia perfectiora accidentia sunt propinquiora substantiae, et eis mediantibus alia accidentia substantiae insunt. In order to understand this third division, one must consider that, since relation has the weakest existence—consisting alone, as it does, in the fact of being something referred to something else—it is necessary that it be grounded on some other accident. For the more perfect accidents are closer to the substance, and it is through them as intermediates that the other accidents inhere in the substance. Maxime autem super duo fundatur relatio, quae habent ordinem ad aliud, scilicet super quantitatem et actionem: nam quantitas potest esse mensura etiam alicuius exterioris; agens autem transfundit actionem suam in aliud. Now relation is founded chiefly upon two accidents which have an order to something else, namely, upon quantity and action. [or quantity may be a measure even of something external to it; while the agent transfuses its action into something other than itself. Relationes igitur quaedam fundantur super quantitatem; et praecipue super numerum, cui competit prima ratio mensurae, ut patet in duplo et dimidio, multiplici et submultiplici, et in aliis huiusmodi. Idem etiam et simile et aequale fundantur super unitatem, quae est principium numeri. Accordingly, certain relations are founded upon quantity; and especially upon that species of quantity which is number, to which the basic notion of measure pertains, as—is evident in “double and half,” “multiple and submultiple [fraction]” and other such. Similarly, “same,” “like,” and “equal” are founded upon unity, which is the principle of number. Aliae vero relationes fundantur super actionem et passionem: vel secundum ipsum actum, sicut calefaciens dicitur ad calefactum; vel secundum hoc quod est egisse, sicut pater refertur ad filium quia genuit; vel secundum potentiam agendi, sicut dominus ad servum quia potest eum coercere. Still other relations are founded upon action and passion: either according to existing act [in the present], as something is said to be “heating” in relation to that which is heated; or according to “having acted” [in the past], as a father is referred to a son because he because he engendered him; or else according to the possibility of acting [in the future], as master is related to a servant because he is able to make him do something. Hanc igitur divisionem manifeste expressit philosophus in V Metaphys.; sed hic breviter tangit, dicens quod ad aliquid aliud quidem est secundum superabundantiam et defectum; quod quidem fundatur super quantitatem, ut duplum et dimidium: aliud autem secundum activum et passivum, et motivum et mobile, quae ad invicem referuntur, ut patet per se. Now the Philosopher clearly explains this division in Metaphysics V (l.17); but he here touches on it briefly, saying that one sort of “to something” [relation] is that according to “excess and defect,” which sort, indeed, is founded on quantity, as in the case of “double and half”; while the other is according to “active and passive,” and “mover and movable,” which are referred to each other, as is self-evident. Deinde cum dicit: non est autem motus praeter res etc., ostendit quomodo motus reducitur ad praedictas divisiones. 281. Then  he shows how motion is reduced to the aforesaid [three] divisions. Et circa hoc duo facit: primo ostendit quod motus non est praeter genera rerum in quibus contingit esse motum; secundo quod dividitur sicut genera rerum dividuntur, ibi: unumquodque autem et cetera. And as to this he does two things: He shows that motion is not outside the genera of things in which motion occurs; He shows that motion is divided as the genera of things are divided, at 282. Circa primum considerandum est quod, cum motus, sicut infra patebit, sit actus imperfectus; omne autem quod est imperfectum, sub eodem genere cadit cum perfecto, non quidem sicut species, sed per reductionem (sicut materia prima est in genere substantiae); necesse est quod motus non sit praeter genera rerum in quibus contingit esse motum. Et hoc est quod dicit, quod motus non est praeter res, idest praeter genera rerum in quibus est motus, ita quod sit aliquid extraneum, vel aliquid commune ad haec genera. As to the first of these, it should be observed that since motion, as will be evident below (lesson following, nos. 285, 287; l.3, no. 296), is an imperfect act, and since everything which is imperfect falls under the same genus with that which is perfect in respect to it—not, indeed, as a species, but by reduction (as prime matter is by reduction in the genus of “substance”) necessarily motion is not outside the genera of things in which motion occurs. And this is what he [Aristotle] states, namely, that motion is not “outside of things,” i.e., outside the genera of things in which motion is found, in such a way as to be something extraneous to, or something common to, these genera. Et hoc manifestat per hoc quod omne quod mutatur, mutatur vel secundum substantiam, vel secundum quantitatem, vel secundum qualitatem, vel secundum locum, ut in quinto ostendetur. And he makes this plain by the fact that whatever is changed, is changed either according to substance, or quantity, or quality, or place, as will be shown in Book V. His autem generibus non est accipere aliquod commune univocum, quod non contineatur sub aliquo praedicamento, sed sit genus eorum: sed ens est commune ad ea secundum analogiam, ut in IV Metaphys. ostendetur. Unde etiam manifestum est quod neque motus neque mutatio sunt extra praedicta genera; cum nihil sit extra ea, sed sufficienter dividant ens. Quomodo autem motus se habeat ad praedicamentum actionis vel passionis, infra ostendetur. Now there is not to be found in these genera some common univocal element which would not be found under some predicament but would be their genus; but being is common to them according to analogy, as will be shown in Metaphysics IV (l.1). Whence it is also plain that neither motion nor change is outside the aforesaid genera, since nothing is outside the latter and they sufficiently divide being. But, as to the question of how motion is related to the predicament of action or passion, this will be explained below (l.5). Deinde cum dicit: unumquodque autem dupliciter etc., ostendit quod motus dividitur sicut genera rerum. 282. Then  he shows that motion is divided as the genera of things are divided, Manifestum est enim quod in omnibus generibus contingit aliquid esse dupliciter, vel sicut perfectum, vel sicut imperfectum. Cuius ratio est, quia privatio et habitus est prima contrarietas, quae in omnibus contrariis salvatur, ut in X Metaphys. dicitur. Unde, cum omnia genera dividantur contrariis differentiis, oportet in omnibus generibus esse perfectum et imperfectum: sicut in substantia aliquid est ut forma, et aliquid ut privatio; et in qualitate aliquid est ut album quod est perfectum, et aliquid ut nigrum, quod est quasi imperfectum; et in quantitate, aliquid est quantitas perfecta et aliquid imperfecta; et in loco aliquid est sursum, quod est quasi perfectum, et aliquid deorsum, quod est quasi imperfectum; vel leve et grave, quae ponuntur in ubi, ratione inclinationis. Unde manifestum est quod quot modis dividitur ens, tot modis dividitur motus. For it is plain that in all the genera a thing may be present in two ways, either as something perfect, or as something imperfect. The reason for this is that privation and possession is the prime contrariety, which is found in all the contraries, as is stated in Metaphysics X (l.6). Whence, since all the genera are divided through contrary differences, it is necessary that in all there be the perfect and the imperfect: as in “substance” something is as form and something is as privation; and in “quality” there is something such as white, which is perfect, and something such as black, which is, as it were, imperfect; and in “quantity” one thing is perfect quantity, another imperfect; and in “place” something is above, which is, as it were, perfect, and something is below, which is, so to speak, imperfect; or else there is light and heavy, which are placed in “where” [place] by virtue of the inclination [to a certain place] which is in them. Hence it is plain that according to the divisions of being, there are corresponding divisions of motion. Differunt enim species motus secundum diversa genera entium; ut augmentum, quod est motus in quantitate, a generatione, quae est motus in substantia. For the species of motion differ according to the different genera of being—as “increase,” which is motion in quantity, differs from “generation,” which is motion in substance. Differunt etiam species motus secundum perfectum et imperfectum in eodem genere: nam generatio est motus in substantia ad formam, corruptio vero ad privationem; et in quantitate augmentum ad quantitatem perfectam, diminutio ad imperfectam. Quare autem non assignentur duae species in qualitate et in ubi ostendetur in quinto. The species of motion likewise differ according to perfect and imperfect in the same genus: for “generation” is motion in substance toward form, while “corruption” is motion toward privation; and in quantity, “increase” is toward perfect quantity, “diminution” toward imperfect. But as to the question of why there are not assigned two kinds in quality and where [place], this will be explained in Book V (l.4).
Definition of Motion
Chapter 1 cont. διῃρημένου δὲ καθ' ἕκαστον γένος τοῦ μὲν ἐντελεχείᾳ τοῦ δὲ δυνάμει, ἡ τοῦ δυνάμει ὄντος ἐντελέχεια, ᾗ τοιοῦτον, κίνησίς ἐστιν, We have now before us the distinctions in the various classes of being between what is full real and what is potential. Def. The fulfilment of what exists potentially, in so far as it exists potentially, is motion, οἷον τοῦ μὲν ἀλλοιωτοῦ, ᾗ ἀλλοιωτόν, ἀλλοίωσις, τοῦ δὲ αὐξητοῦ καὶ τοῦ ἀντικειμένου φθιτοῦ (οὐδὲν γὰρ ὄνομα κοινὸν ἐπ' ἀμφοῖν) αὔξησις καὶ φθίσις, τοῦ δὲ γενητοῦ καὶ φθαρτοῦ γένεσις καὶ φθορά, τοῦ δὲ φορητοῦ φορά. namely, of what is alterable qua alterable, alteration: of what can be increased and its opposite what can be decreased (there is no common name), increase and decrease: of what can come to be and can pass away, coming to he and passing away: of what can be carried along, locomotion. ὅτι δὲ τοῦτο ἔστιν ἡ κίνησις, ἐντεῦθεν δῆλον. ὅταν γὰρ τὸ οἰκοδομητόν, ᾗ τοιοῦτον αὐτὸ λέγομεν εἶναι, ἐντελεχείᾳ ᾖ, οἰκοδομεῖται, καὶ ἔστιν τοῦτο οἰκοδόμησις· ὁμοίως δὲ καὶ μάθησις καὶ ἰάτρευσις καὶ κύλισις καὶ ἅλσις καὶ ἅδρυνσις καὶ γήρανσις. Examples will elucidate this definition of motion. When the buildable, in so far as it is just that, is fully real, it is being built, and this is building. Similarly, learning, doctoring, rolling, leaping, ripening, aging. ἐπεὶ δ' ἔνια ταὐτὰ καὶ δυνάμει καὶ ἐντελεχείᾳ ἐστίν, οὐχ ἅμα δὲ ἢ οὐ κατὰ τὸ αὐτό, ἀλλ' οἷον θερμὸν μὲν ἐντελεχείᾳ ψυχρὸν δὲ δυνάμει, πολλὰ ἤδη ποιήσει καὶ πείσεται ὑπ' ἀλλήλων· ἅπαν γὰρ ἔσται ἅμα ποιητικὸν καὶ παθητικόν. ὥστε καὶ τὸ κινοῦν φυσικῶς κινητόν· πᾶν γὰρ τὸ τοιοῦτον κινεῖ κινούμενον καὶ αὐτό. δοκεῖ μὲν οὖν τισιν ἅπαν κινεῖσθαι τὸ κινοῦν, οὐ μὴν ἀλλὰ περὶ τούτου μὲν ἐξ ἄλλων ἔσται δῆλον ὅπως ἔχει (ἔστι γάρ τι κινοῦν καὶ ἀκίνητον), ἡ δὲ τοῦ δυνάμει ὄντος <�ἐντελέχεια>, ὅταν ἐντελεχείᾳ ὂν ἐνεργῇ οὐχ ᾗ αὐτὸ ἀλλ' ᾗ κινητόν, κίνησίς ἐστιν. The same thing, if it is of a certain kind, can be both potential and fully real, not indeed at the same time or not in the same respect, but e.g. potentially hot and actually cold. Hence at once such things will act and be acted on by one another in many ways: each of them will be capable at the same time of causing alteration and of being altered. Hence, too, what effects motion as a physical agent can be moved: when a thing of this kind causes motion, it is itself also moved. This, indeed, has led some people to suppose that every mover is moved. But this question depends on another set of arguments, and the truth will be made clear later. is possible for a thing to cause motion, though it is itself incapable of being moved. It is the fulfilment of what is potential when it is already fully real and operates not as itself but as movable, that is motion. λέγω δὲ τὸ ᾗ ὡδί. ἔστι γὰρ ὁ χαλκὸς δυνάμει ἀνδριάς, ἀλλ' ὅμως οὐχ ἡ τοῦ χαλκοῦ ἐντελέχεια, ᾗ χαλκός, κίνησίς ἐστιν· οὐ γὰρ τὸ αὐτὸ τὸ χαλκῷ εἶναι καὶ δυνάμει τινί [κινητῷ], ἐπεὶ εἰ ταὐτὸν ἦν ἁπλῶς καὶ κατὰ τὸν λόγον, ἦν ἂν ἡ τοῦ χαλκοῦ, ᾗ χαλκός, ἐντελέχεια κίνησις· οὐκ ἔστιν δὲ ταὐτόν, ὡς εἴρηται What I mean by 'as' is this: Bronze is potentially a statue. But it is not the fulfilment of bronze as bronze which is motion. For 'to be bronze' and 'to be a certain potentiality' are not the same. If they were identical without qualification, i.e. in definition, the fulfilment of bronze as bronze would have been motion. But they are not the same, as has been said. (δῆλον δ' ἐπὶ τῶν ἐναντίων· τὸ μὲν γὰρ δύνασθαι ὑγιαίνειν (201b.) καὶ δύνασθαι κάμνειν ἕτερον—καὶ γὰρ ἂν τὸ κάμνειν καὶ τὸ ὑγιαίνειν ταὐτὸν ἦν—τὸ δὲ ὑποκείμενον καὶ τὸ ὑγιαῖνον καὶ τὸ νοσοῦν, εἴθ' ὑγρότης εἴθ' αἷμα, ταὐτὸν καὶ ἕν). ἐπεὶ δ' οὐ ταὐτόν, ὥσπερ οὐδὲ χρῶμα ταὐτὸν καὶ ὁρατόν, ἡ τοῦ δυνατοῦ, ᾗ δυνατόν, ἐντελέχεια φανερὸν ὅτι κίνησίς ἐστιν. (This is obvious in contraries. 'To be capable of health' and 'to be capable of illness' are not the same, for if they were there would be no difference between being ill and being well. Yet the subject both of health and of sickness—whether it is humour or blood—is one and the same.) We can distinguish, then, between the two—just as, to give another example, 'colour' and visible' are different—and clearly it is the fulfilment of what is potential as potential that is motion. So this, precisely, is motion.
Postquam philosophus praemisit quaedam, quae sunt necessaria ad inquisitionem definitionis motus, hic definit motum: et primo in generali; secundo in speciali, ibi: quid quidem igitur motus et cetera. 283. After first setting down certain things necessary for investigating the definition of motion, the Philosopher now defines motion: In general; More specifically, at 325. Circa primum duo facit: primo ostendit quid sit motus; secundo inquirit cuius actus sit motus, utrum moventis aut mobilis, ibi: movetur autem movens et cetera. With regard to the first, he does two things: He shows what motion is; He inquires whether motion belongs to the mover or to the mobile thing, at 299. Circa primum tria facit: primo ponit definitionem motus; secundo manifestat partes definitionis, ibi: quod autem hoc sit motus etc.; tertio ostendit definitionem esse bene assignatam, ibi: quod quidem igitur hoc sit et cetera. As to the first of these, he does three things: He gives the definition of motion; He explains the parts of the definition, at 287. He shows that it is a good definition, at 291. Circa primum duo facit: primo ponit definitionem motus; secundo exemplificat, ibi: ut alterabilis quidem et cetera. As to the first, he does two things: He gives the definition of motion; He gives examples, at 286. Circa primum sciendum est, quod aliqui definierunt motum dicentes, quod motus est exitus de potentia in actum non subito. Qui in definiendo errasse inveniuntur, eo quod in definitione motus posuerunt quaedam quae sunt posteriora motu: exitus enim est quaedam species motus; subitum etiam in sua definitione recipit tempus: est enim subitum, quod fit in indivisibili temporis; tempus autem definitur per motum. 284. As to the first, one must understand that some have defined motion by saying that motion is “a going-out from potency to act which is not sudden.” But they are found to be in error, because they have placed in the definition certain elements that are posterior to motion: for a “going-out” is a species of motion; “sudden,” likewise, involves time in its definition—the “sudden” is that which occurs in the indivisible of time [i.e., the instant]; time, however, is defined in terms of motion. Et ideo omnino impossibile est aliter definire motum per priora et notiora, nisi sicut philosophus hic definit. Dictum est enim quod unumquodque genus dividitur per potentiam et actum. Potentia autem et actus, cum sint de primis differentiis entis, naturaliter priora sunt motu: et his utitur philosophus ad definiendum motum. 285. Consequently it is entirely impossible to define motion in terms of what is prior and better known otherwise than the Philosopher here does. For it has been pointed out already that every genus is divided by potency and act. Now potency and act, since they are among the first differences of being, are naturally prior to motion, and it is these that the Philosopher uses to define motion. Considerandum est igitur quod aliquid est in actu tantum, aliquid vero in potentia tantum, aliquid vero medio modo se habens inter potentiam et actum. Quod igitur est in potentia tantum, nondum movetur: quod autem iam est in actu perfecto, non movetur, sed iam motum est: illud igitur movetur, quod medio modo se habet inter puram potentiam et actum, quod quidem partim est in potentia et partim in actu; ut patet in alteratione. Cum enim aqua est solum in potentia calida, nondum movetur: cum vero est iam calefacta, terminatus est motus calefactionis: cum vero iam participat aliquid de calore sed imperfecte, tunc movetur ad calorem; nam quod calefit, paulatim participat calorem magis ac magis. Ipse igitur actus imperfectus caloris in calefactibili existens, est motus: non quidem secundum id quod actu tantum est, sed secundum quod iam in actu existens habet ordinem in ulteriorem actum; quia si tolleretur ordo ad ulteriorem actum, ipse actus quantumcumque imperfectus, esset terminus motus et non motus, sicut accidit cum aliquid semiplene calefit. Ordo autem ad ulteriorem actum competit existenti in potentia ad ipsum. Consider, therefore, that something is in act only, something is in potency only, something else is midway between potency and act. What is in potency only is not yet being moved; what is already in perfect act is not being moved but has already been moved. Consequently, that is being moved which is midway between pure potency and act, which is partly in potency and partly in act—as is evident in alteration. [or when water is only potentially hot, it is not being moved; when it has now been heated, the motion of heating is finished; but when it possesses “ some heat, through imperfectly, then it is being moved—for whatever is being heated gradually acquires heat step by step. Therefore this imperfect act of heat existing in a heatable object is motion—not, indeed, by reason of what the heatable object has already become, but inasmuch as, being already in act, it has an order to a further act. For should this order to a further act be taken away, the act already present, however, imperfect, would be the term of motion and not motion itself—as happens when something becomes half-heated. This order to a further act belongs to the thing that is in potency to it. Et similiter, si actus imperfectus consideretur tantum ut in ordine ad ulteriorem actum, secundum quod habet rationem potentiae, non habet rationem motus, sed principii motus: potest enim incipere calefactio sicut a frigido, ita et a tepido. Similarly, if the imperfect act were considered solely as ordered to a further act, under its aspect of potency, it would not have the nature of motion but of a principle of motion—for heating can begin from either a cold or a lukewarm object. Sic igitur actus imperfectus habet rationem motus, et secundum quod comparatur ad ulteriorem actum ut potentia, et secundum quod comparatur ad aliquid imperfectius ut actus. The imperfect act, therefore, has the character of motion both insofar as is compared, as potency, to a further act, and insofar as it is compared, as act, to something more imperfect. Unde neque est potentia existentis in potentia, neque est actus existentis in actu, sed est actus existentis in potentia: ut per id quod dicitur actus, designetur ordo eius ad anteriorem potentiam, et per id quod dicitur in potentia existentis, designetur ordo eius ad ulteriorem actum. Hence, motion is neither the potency of a thing existing in potency, nor the act of a thing in act, but it is the act of a thing in potency; where the word “act” designates its relation to a prior potency, and the words “of a thing in potency” designates its relation to a further act. Unde convenientissime philosophus definit motum, dicens quod motus est entelechia, idest actus existentis in potentia secundum quod huiusmodi. Whence the Philosopher most aptly defines motion as the entelechy, i.e., the act, of a thing existing in potency insofar as it is in potency. Deinde cum dicit: ut alterabilis quidem etc., exemplificat in omnibus speciebus motus: sicut alteratio est actus alterabilis inquantum est alterabile. 286. Then  he gives examples from all the species of motion—as, for example, alteration is the act of the alterable insofar as it is alterable. Et quia motus in quantitate et in substantia non habent unum nomen, sicut motus in qualitate dicitur alteratio, quantum ad motum in quantitate ponit duo nomina: et dicit quod actus augmentabilis et oppositi, scilicet diminuibilis, quibus non est unum commune nomen, est augmentum et diminutio. Et similiter generabilis et corruptibilis, generatio et corruptio; et mutabilis secundum locum, loci mutatio. And because motion in quantity and in substance does not have a single name in the same way as motion in quality is called “alteration,” he gives two different names for the motions in quantity, and says that the act of the increasable, and of its opposite, i.e., the decreasable, for which two there is no common name, is “increase” and “decrease.” Similarly, the acts of the generable and of the corruptible are “generation” and “corruption”; and the act of what is mutable in regard to place is called “change of place.” Accipit enim hic motum communiter pro mutatione, non autem stricte secundum quod dividitur contra generationem et corruptionem, ut dicetur in quinto. In this section the Philosopher uses the word “motion” for any kind of change and avoids the strict usage in which “motion” is distinct from “generation” and “corruption,” as will be said in Book V. Deinde cum dicit: quod autem hoc sit motus etc., manifestat singulas particulas definitionis: et primo quantum ad hoc quod motus dicitur actus; secundo quantum ad hoc quod dicitur existentis in potentia, ibi: quoniam autem etc.; tertio quantum ad hoc quod additur, inquantum huiusmodi, ibi: dico autem hoc et cetera. 287. Then  he explains the several words of the definition: He explains the use of the word “act”; He explains “of a thing existing in potency,” at 288. He explains “insofar as it is such,” at 289. Circa primum utitur tali ratione. Id quo aliquid fit actu, prius in potentia existens, est actus; sed aliquid fit actu dum movetur, prius adhuc in potentia existens; ergo motus est actus. As to the first he uses this reasoning. That by which something previously existing in potency becomes actual is an act. But something becomes actual when it is being moved, although previously it was in potency. Therefore motion is an act. Dicit ergo ex hoc manifestum esse quod motus sit hoc, idest actus, quia aedificabile dicit potentiam ad aliquid; cum autem aedificabile secundum hanc potentiam quam importat, reducitur in actum, tunc dicimus quod aedificatur: et iste actus est aedificatio passiva. Et similiter est de omnibus aliis motibus, sicut doctrinatio, medicatio, volutatio, saltatio, adolescentia, idest augmentum, et senectus, idest diminutio. He says therefore that it is plain that motion is an act from the fact that the “buildable” implies a potency to something, but when the “buildable” according to this potency which it implies, is being reduced to act, we then say it is “being built”—and this act is “building” taken passively. And the same thing is true in all other motions such as indoctrination, healing, rolling, jumping, youth (i.e., increase),old age (i.e., decrease). Considerandum est enim quod antequam aliquid moveatur, est in potentia ad duos actus, scilicet ad actum perfectum, qui est terminus motus, et ad actum imperfectum, qui est motus: sicut aqua antequam incipiat calefieri est in potentia ad calefieri et ad calidum esse; cum autem calefit, reducitur in actum imperfectum, qui est motus; nondum autem in actum perfectum, qui est terminus motus, sed adhuc respectu ipsius remanet in potentia. For it must be remembered that before something is being moved it is in potency to two acts: to a perfect act which is the term of the motion, and to an imperfect act which is motion itself. Thus water, before it begins to be heated, is in potency to being heated and to having been heated: when it is being heated it is being reduced to the imperfect act which is motion but not yet to perfect act which is the term of the motion—rather, in respect to this it still remains in potency. Deinde cum dicit: quoniam autem quaedam etc., ostendit quod motus sit actus existentis in potentia, tali ratione. 288. Then [1977 he shows that motion is the act “of a thing existing in potency.” Omnis enim actus, eius est proprie actus in quo semper invenitur, sicut lumen nunquam invenitur nisi in diaphano, et propter hoc est actus diaphani. Sed motus semper invenitur in existente in potentia; est igitur motus actus existentis in potentia. For every act is strictly the act of that in which it is always found—as light is never found but in the transparent, for which reason it is the act of the transparent. But motion is found always in a thing existing in potency. Therefore, motion is the act of a thing existing in potency. Ad manifestationem igitur secundae propositionis, dicit quod quia quaedam eadem sunt et in potentia et in actu, licet non simul aut secundum idem, sicut aliquid est calidum in potentia et frigidum actu; ex hoc sequitur quod multa agunt et patiuntur ad invicem, inquantum scilicet utrumque est in potentia et actu respectu alterius secundum diversa. Et quia omnia corpora naturalia inferiora communicant in materia, ideo in unoquoque est potentia ad id quod est actu in altero: et ideo in omnibus talibus aliquid simul agit et patitur, et movet et movetur. To explain the second proposition he says that, since certain same things are both in potency and in act, although not at the same time, nor in the same respect—as, for example, something is hot actually and cold potentially it follows that many things mutually act and are acted upon insofar, namely, as both are in potency and in act with respect to the other under different aspects. And because all lower natural bodies share the same matter, there is therefore in each of them a potency to what is actual in another. Hence, in all such bodies something simultaneously acts and is acted upon, both moves and is moved. Et ex hac ratione quibusdam visum est quod simpliciter omne movens moveatur: sed de hoc manifestum erit magis in aliis. Ostendetur enim in octavo huius et in XII Metaphys., quod est quoddam movens immobile, quia non est in potentia sed in actu tantum. This fact had led some to say absolutely that every mover is likewise being moved. This point will be cleared up in a later place. For it will be shown in Physics VIII (l.9 ff.) and in Metaphysics XII (l.7) that there exists an immobile mover, since it is not in potency but in act only. Sed quando id quod est in potentia, actu quodammodo existens, agit aut ipsum aut aliud inquantum est mobile, idest reducitur in actum motus, sive sit motum a se sive ab alio, tunc est motus actus eius. Et inde est quod illa quae sunt in potentia, sive agant sive patiantur, moventur; quia et agendo patiuntur, et movendo moventur: sicut ignis, cum agit in ligna, patitur inquantum ingrossatur per fumum, quia flamma non est nisi fumus ardens. But when that which is in potency, yet existing in a certain way in act, either acts itself or is acted upon by another so far as it is movable, i.e., is reduced to the act of motion, whether moved by itself or by another, at such time motion is its act. That is why things in potency, whether they act or are acted upon, are moved, since when acting they are acted upon and when moving they are being moved—just as fire, when it acts on logs, it acted upon, insofar as it becomes more dense through smoke, flame being nothing more than smoke afire. Deinde cum dicit: dico autem hoc inquantum etc., manifestat hanc particulam, inquantum huiusmodi: et primo per exemplum; secundo per rationem, ibi: manifestum autem et in contrariis et cetera. 289. Then  he explains this part of the definition, “insofar as it is such”: By an example; By giving a reason, at 290. Dicit ergo primo quod necessarium fuit addi inquantum huiusmodi, quia id quod est in potentia, est etiam aliquid actu. Et licet idem sit subiectum existens in potentia et in actu, non tamen est idem secundum rationem esse in potentia et esse in actu, sicut aes est in potentia ad statuam et est actu aes, non tamen est eadem ratio aeris inquantum est aes et inquantum est potentia ad statuam. Motus autem non est actus aeris inquantum est aes, sed inquantum est in potentia ad statuam: alias oporteret quod quandiu aes esset, tandiu aes moveretur, quod patet esse falsum. Unde patet convenienter additum esse inquantum huiusmodi. He says therefore first that the phrase, “insofar as it is such,” had to be added, because what is in potency is at the same time something in act. And although the subject which is both in potency and in act may be the same, nevertheless to be in potency and to be in act is not contained under the same notion. Thus, although brass is a statue in potency but is brass actually, nevertheless the notion of the brass as brass is not the same as the notion of the brass as it is in potency to a statue. Now motion is not an act of the brass insofar as it is brass but insofar as it is in potency to a statue; otherwise. during the whole time that it was brass it would be undergoing motion, which is clearly false. That is why it is necessary to add “insofar as it is such. Deinde cum dicit: manifestum autem et in contrariis etc., ostendit idem per rationem sumptam a contrariis. Manifestum est enim quod aliquod idem subiectum est in potentia ad contraria, sicut humor aut sanguis est idem subiectum se habens in potentia ad sanitatem et aegritudinem. Manifestum est autem quod esse in potentia ad sanitatem, et esse in potentia ad aegritudinem, est alterum et alterum (et hoc dico secundum ordinem ad obiecta): alioquin si idem esset posse laborare et posse sanari, sequeretur quod laborare et sanari essent idem. Differunt ergo posse laborare et posse sanari secundum rationem, sed subiectum est unum et idem. 290. Then  he explains the same thing by using an argument based on the nature of contraries. For it is clear that a given subject is in potency to contraries—as a humor or the blood is in potency to health and to sickness. But to be in potency to health is one thing and to be in potency to sickness is another, if one considers their objects. Otherwise, if to be able to be sick and to be able to be well were the same thing, it then would follow that being sick and being well would be the same. Hence to be able to be sick and to be able to be healthy are different notions, although their actual subject is one and the same thing. Patet ergo quod non est eadem ratio subiecti inquantum est quoddam ens, et inquantum est potentia ad aliud: alioquin potentia ad contraria esset una secundum rationem. Et sic etiam non est idem secundum rationem color et visibile. It is plain, therefore, that there is not one and the same notion of the subject as it is a certain being, and as it is in potency to something else. Otherwise, potency to contrary things would fall under one and the same notion. In like manner, the notion of that which is “color” and that which is “visible” are not one and the same. Et ideo necessarium fuit dicere quod motus est actus possibilis inquantum est possibile: ne intelligeretur esse actus eius quod est in potentia, inquantum est quoddam subiectum. Thus it was necessary to say that motion is the act of the possible “insofar as it is possible”—to prevent supposing that it is the act of what is in potency insofar as it is merely some subject.
Justification of the definition of motion
Chapter 1 cont. ὅτι μὲν οὖν ἐστιν αὕτη, καὶ ὅτι συμβαίνει τότε κινεῖσθαι ὅταν ἡ ἐντελέχεια ᾖ αὐτή, καὶ οὔτε πρότερον οὔτε ὕστερον, δῆλον· ἐνδέχεται γὰρ ἕκαστον ὁτὲ μὲν ἐνεργεῖν ὁτὲ δὲ μή, οἷον τὸ οἰκοδομητόν, καὶ ἡ τοῦ οἰκοδομητοῦ ἐνέργεια, ᾗ οἰκοδομητόν, οἰκοδόμησίς ἐστιν (ἢ γὰρ οἰκοδόμησις ἡ ἐνέργεια [τοῦ οἰκοδομητοῦ] ἢ ἡ οἰκία· ἀλλ' ὅταν οἰκία ᾖ, οὐκέτ' οἰκοδομητὸν ἔστιν· οἰκοδομεῖται δὲ τὸ οἰκοδομητόν· ἀνάγκη οὖν οἰκοδόμησιν τὴν ἐνέργειαν εἶναι)· ἡ δ' οἰκοδόμησις κίνησίς τις. ἀλλὰ μὴν ὁ αὐτὸς ἐφαρμόσει λόγος καὶ ἐπὶ τῶν ἄλλων κινήσεων. Further it is evident that motion is an attribute of a thing just when it is fully real in this way, and neither before nor after. For each thing of this kind is capable of being at one time actual, at another not. Take for instance the buildable as buildable. The actuality of the buildable as buildable is the process of building. For the actuality of the buildable must be either this or the house. But when there is a house, the buildable is no longer buildable. On the other hand, it is the buildable which is being built. The process then of being built must be the kind of actuality required But building is a kind of motion, and the same account will apply to the other kinds also. Chapter 2 Ὅτι δὲ καλῶς εἴρηται, δῆλον καὶ ἐξ ὧν οἱ ἄλλοι περὶ αὐτῆς λέγουσιν, καὶ ἐκ τοῦ μὴ ῥᾴδιον εἶναι διορίσαι ἄλλως αὐτήν. οὔτε γὰρ τὴν κίνησιν καὶ τὴν μεταβολὴν ἐν ἄλλῳ γένει θεῖναι δύναιτ' ἄν τις, The soundness of this definition is evident both when we consider the accounts of motion that the others have given, and also from the difficulty of defining it otherwise. One could not easily put motion and change in another genus. δῆλόν τε σκοποῦσιν ὡς τιθέασιν αὐτὴν ἔνιοι, ἑτερότητα καὶ ἀνισότητα καὶ τὸ μὴ ὂν φάσκοντες εἶναι τὴν κίνησιν· ὧν οὐδὲν ἀναγκαῖον κινεῖσθαι, οὔτ' ἂν ἕτερα ᾖ οὔτ' ἂν ἄνισα οὔτ' ἂν οὐκ ὄντα· ἀλλ' οὐδ' ἡ μεταβολὴ οὔτ' εἰς ταῦτα οὔτ' ἐκ τούτων μᾶλλόν ἐστιν ἢ ἐκ τῶν ἀντικειμένων. This is plain if we consider where some people put it; they identify motion with or 'inequality' or 'not being'; but such things are not necessarily moved, whether they are 'different' or 'unequal' or 'non-existent'; Nor is change either to or from these rather than to or from their opposites. αἴτιον δὲ τοῦ εἰς ταῦτα τιθέναι ὅτι ἀόριστόν τι δοκεῖ εἶναι ἡ κίνησις, τῆς δὲ ἑτέρας συστοιχίας αἱ ἀρχαὶ διὰ τὸ στερητικαὶ εἶναι ἀόριστοι· οὔτε γὰρ τόδε οὔτε τοιόνδε οὐδεμία αὐτῶν ἐστιν, [ὅτι] οὐδὲ τῶν ἄλλων κατηγοριῶν. The reason why they put motion into these genera is that it is thought to be something indefinite, and the principles in the second column are indefinite because they are privative: none of them is either 'this' or 'such' or comes under any of the other modes of predication. τοῦ δὲ δοκεῖν ἀόριστον εἶναι τὴν κίνησιν αἴτιον ὅτι οὔτε εἰς δύναμιν τῶν ὄντων οὔτε εἰς ἐνέργειαν ἔστιν θεῖναι αὐτήν· οὔτε γὰρ τὸ δυνατὸν ποσὸν εἶναι κινεῖται ἐξ ἀνάγκης οὔτε τὸ ἐνεργείᾳ ποσόν, ἥ τε κίνησις ἐνέργεια μὲν εἶναί τις δοκεῖ, ἀτελὴς δέ· αἴτιον δ' ὅτι ἀτελὲς τὸ δυνατόν, οὗ ἐστιν ἐνέργεια. καὶ διὰ τοῦτο δὴ χαλεπὸν αὐτὴν λαβεῖν τί ἐστιν· ἢ γὰρ εἰς στέρησιν ἀναγκαῖον θεῖναι ἢ εἰς δύναμιν ἢ εἰς ἐνέργειαν ἁπλῆν, τούτων δ' οὐδὲν φαίνεται ἐνδεχόμενον. λείπεται (202a.) τοίνυν ὁ εἰρημένος τρόπος, ἐνέργειαν μέν τινα εἶναι, τοιαύτην δ' ἐνέργειαν οἵαν εἴπαμεν, χαλεπὴν μὲν ἰδεῖν, ἐνδεχομένην δ' εἶναι. The reason in turn why motion is thought to be indefinite is that it cannot be classed simply as a potentiality or as an actuality—a thing that is merely capable of having a certain size is not undergoing change, nor yet a thing that is actually of a certain size, and motion is thought to be a sort of actuality, but incomplete, the reason for this view being that the potential whose actuality it is is incomplete. This is why it is hard to grasp what motion is. It is necessary to class it with privation or with potentiality or with sheer actuality, yet none of these seems possible. There remains then the suggested mode of definition, namely that it is a sort of actuality, or actuality of the kind described, hard to grasp, but not incapable of existing.
Posita definitione motus et manifestatis singulis particulis definitionis, hic consequenter ostendit quod definitio sit bene assignata: et primo directe; secundo indirecte, ibi: quod autem bene dictum sit et cetera. 291. Having given the definition of motion and an explanation of each of the words in the definition, the Philosopher now shows it to be a good definition: Directly; Indirectly, at 293. Circa primum utitur tali ratione. Omne quod est in potentia, quandoque contingit esse in actu; sed aedificabile est in potentia; ergo contingit aliquem actum esse aedificabilis inquantum est aedificabile. 292. In regard to the first  he uses the following argument: Everything which is in potency may at some time be in act; butt he “buildable” is in potency. Therefore, there may. at some time be an act of the “buildable” insofar as it is buildable. Hoc autem est vel domus vel aedificatio. Sed domus non est actus aedificabilis inquantum est aedificabile, quia aedificabile inquantum huiusmodi reducitur in actum cum aedificatur; cum autem iam domus est, non aedificatur. Relinquitur igitur quod aedificatio sit actus aedificabilis inquantum huiusmodi. Aedificatio autem est quidam motus: motus igitur est actus existentis in potentia inquantum huiusmodi. Et eadem ratio est de aliis motibus. But this act is either the house itself or the building of it. But “house” is not the act of the “buildable” insofar as it is “buildable.” Since the “buildable” as such is being reduced into act when the building is taking place, but when the house now exists, it is no longer being built. Hence, building is an act of the buildable as such. Building, however, is a certain motion. Motion, therefore, is the act of a thing existing in potency as such. The same is true of other motions. Patet igitur quod motus sit talis actus qualis dictus est et quod tunc aliquid movetur, quando est in tali actu, et neque prius neque posterius: quia prius, cum est in potentia tantum, non incipit motus; neque etiam posterius, cum iam omnino desinat esse in potentia per hoc quod sit in actu perfecto. It is clear, therefore, that motion is the type of act above-described, and that something is being moved only when it is in such an act, and neither before nor after—not before, since if it is only in potency the motion has not begun; nor after, since it has now completely ceased to be in potency, by virtue of being in perfect act. Deinde cum dicit: quod autem bene dictum sit etc., ostendit indirecte definitionem esse bene assignatam, per hoc scilicet quod non contingit motum aliter definire. 293. Then  he shows indirectly that it is a good definition by showing that motion cannot be defined in any other way. In regard to this he does three things: Et circa hoc tria facit: primo proponit quod intendit; secundo ponit definitiones aliorum de motu, et reprobat eas, ibi: manifestum autem intendentibus etc.; tertio assignat causam quare alii sic definierunt motum, ibi: causa autem in hoc ponere et cetera. He proposes what he intends; He presents definitions given by others and rejects them, at 294. He explains why others defined motion as they did, at 295. Dicit ergo primo quod manifestum est motum esse bene definitum ex duobus: primo quidem quia definitiones quibus alii definierunt motum, sunt inconvenientes; secundo ex hoc quod non contingit eum aliter definire. Cuius ratio est, quia motus non collocari potest in aliquo alio genere quam in genere actus existentis in potentia. He says therefore that two things show why the definition given of motion is a good one: First, because the definitions that others have given are unsuitable; Secondly, because it is impossible to define motion otherwise than as Aristotle has defined it, the reason being that motion cannot be placed in any other genus but that of “act of a thing existing in potency.” Deinde cum dicit: manifestum autem intendentibus etc., excludit definitiones aliorum de motu. Et sciendum est quod tripliciter aliqui definierunt motum. Dixerunt enim motum esse alteritatem, propter hoc quod id quod movetur semper alio et alio modo se habet. Item dixerunt motum esse inaequalitatem, quia id quod movetur semper magis ac magis accedit ad terminum. Dixerunt etiam motum esse quod non est, idest non ens: quia id quod movetur, dum movetur, nondum habet id ad quod movetur; ut quod movetur ad albedinem, nondum est album. 294. Then  he excludes the definitions of motion given by others. These followed a three-fold course in their definitions. For they said that motion is “otherness,” because the thing being moved constantly changes from one state to another. Similarly, they said motion is “unequalness,” because the thing being moved approaches its term always more and more. They also said that motion is “non-being,” because the thing being moved does not yet have that to which it is being moved as long as it is being moved—as that which is being moved toward whiteness is not yet white. Has autem definitiones destruit philosophus tripliciter. These definitions the Philosopher destroys in three ways. Primo quidem ex parte subiecti motus. Si enim motus esset alteritas vel inaequalitas vel non ens, cuicumque ista inessent, necessario moveretur; quia cuicumque inest motus, illud movetur. Sed non est necessarium moveri neque ea quae sunt altera, ex hoc ipso quod altera sunt; neque inaequalia, neque ea quae non sunt. Relinquitur igitur quod alteritas et inaequalitas et non ens, non est motus. He does so first by looking at the subject of motion. For if motion were “otherness” or “unequalness” or “non-being,” then whatever would possess any one of these three characteristics would of necessity be undergoing motion—in whatever this motion is, that thing is being moved. But things that are other are not necessarily being moved by the fact that they are “other,” nor by the fact that they are “unequal,” nor by the fact that they “do not exist.” It follows, therefore, that otherness and unequalness and non-being are not motion. Secundo ostendit idem ex parte termini ad quem: quia motus et mutatio non est magis in alteritatem quam in similitudinem, neque magis in inaequalitatem quam in aequalitatem, neque magis in non esse quam in esse. Nam generatio est mutatio ad esse, et corruptio ad non esse. Non igitur motus magis est alteritas quam similitudo, vel inaequalitas quam aequalitas, vel non ens quam ens. Secondly, he shows the same thing by looking at the term to which the motion is tending for motion and change do not tend more to “otherness” than to “likeness,” or to “unequalness” more than to “equality,” or to “non-being” more than to “being.” For generation is a change to “being”, and corruption to “non-being.” Hence motion is not “otherness” any more than “a likeness,” “unequalness” any more than “equalness,” “non-being” any more than “being.” Tertio ostendit idem ex parte termini a quo: quia sicut motus aliquis est ex alteritate et ex inaequalitate et ex non ente, ita est ex oppositis horum. Non igitur motus magis debet poni in his generibus, quam in oppositis. Thirdly, he shows the same thing by looking at the term from which the motion begins since just as some motions start from otherness and from unequalness and from non-being, so others start from their opposites. Hence there is no reason to place motion in the afore-mentioned genera any more than in their opposites. Deinde cum dicit: causa autem in hoc ponere etc., assignat causam quare praedicto modo antiqui motum definierunt. 295. Then  he points out why some defined motion in the aforesaid ways. In regard to this he does two things: Et circa hoc duo facit: primo assignat causam eius quod dictum est; secundo assignat causam cuiusdam quod supposuerat, ibi: videri autem indeterminatum et cetera. First he assigns the reason of what has already been stated. Secondly, he explains a supposition he had made, at 296. Dicit ergo primo quod ista est causa quare antiqui posuerunt motum in praedictis generibus (scilicet alteritatis, inaequalitatis et non entis), quia motus videtur esse quoddam indeterminatum, idest incompletum et imperfectum, quasi non habens determinatam naturam. Et quia indeterminatus est, propter hoc videtur esse ponendus in genere privativorum. Nam cum Pythagoras poneret duas ordinationes rerum, in quarum utraque ponebat quaedam decem principia; principia quae erant in secunda ordinatione, dicebantur ab ipso indeterminata, quia sunt privativa. Non enim determinantur per formam quae sit in genere substantiae, neque per formam qualitatis, neque per formam aliquam specialem in aliquo horum generum existentem, neque etiam per formam alicuius aliorum praedicamentorum. He says therefore that the reason why the older philosophers placed motion in the above-mentioned genera (namely, “otherness,” “unequalness” and “non-being”) is that motion seems to be something indeterminate, i.e., something incomplete and imperfect as though possessing no determinate nature. And because it is indeterminate, its proper place seemed to be in the genus of privation. For when Pythagoras laid down two ordinations of reality, in each of which he placed ten principles, the principles in the second group were said by him to be indeterminate because they were privative. They were not, indeed, determined by a form in the genus of substance, nor by the form of quality, or by any special form in either of these genera or by the form of any of the other predicaments. In una autem ordinatione ponebant Pythagorici haec decem; scilicet finitum, impar, unum, dexterum, masculum, quietem, rectum, lumen, bonum, triangulum aequilaterum: in alia autem, infinitum, par, multitudinem, sinistrum, feminam, motum, obliquum, tenebram, malum, altera parte longius. In one of these groups the Pythagoreans placed ten things: finite, unequal, one, right, male, rest, straight, light, good, equilateral triangle; in the other they placed: infinite, equal, many, left, female, motion, oblique, dark, evil and scalene triangle. Deinde cum dicit: videri autem indeterminatum etc., assignat causam quare motus ponitur inter indeterminata. Et dicit quod huius causa est, quia neque potest poni sub potentia neque sub actu. Si enim poneretur sub potentia, quidquid esset in potentia, puta ad quantitatem, moveretur secundum quantitatem: et si contineretur sub actu, quidquid esset actu quantum, moveretur secundum quantitatem. 296. Then  he gives the reason why motion is placed among the indeterminates. And he says that the reason for this is that motion cannot be placed either in potency or in act. For if it were placed under potency, whatever would be in potency, for example, to quantity, would be being moved according to quantity. If, on the other hand, it were included under act, then whatever things were actually quantified would be being moved according to quantity. Et quidem verum est quod motus est actus: sed est actus imperfectus, medius inter potentiam et actum. Et quod sit actus imperfectus ex hoc patet, quod illud cuius est actus, est ens in potentia, ut supra dictum est. Et ideo difficile est accipere quid sit motus. Videtur enim in primo aspectu quod vel sit simpliciter actus, vel simpliciter potentia, vel quod contineatur sub privatione, sicut antiqui posuerunt ipsum contineri sub non ente et sub inaequalitate. Sed nullum horum est possibile, ut supra ostensum est: unde relinquitur solus praedictus modus ad definiendum motum; ut scilicet motus sit actus talis qualem diximus, scilicet existentis in potentia. Now it is indeed true that motion is act, yet it is imperfect act, a medium between potency and act. And that it is imperfect act is clear from the fact that that of which it is an act is a being in potency as stated above (l.2, no.285). And that is why it is difficult to grasp what motion is. For at first sight It seems to be either entirely act or entirely potency or else to be contained under privation as it seemed to the ancients who called it “non-being” or “unequalness.” But none of these is possible, as we have shown above (no.294). Hence it follows that there is just one way to define motion; namely, that it is the kind of act we have said, i.e., that of a thing existing in potency. Talem autem actum considerare difficile est propter permixtionem actus et potentiae: tamen esse talem actum non est impossibile, sed contingens. It is difficult to dwell on such an act on account of the commingling of act and potency; yet that there should be such an act is not impossible, but contingent.
Action and passion are the same motion
Chapter 2 cont. κινεῖται δὲ καὶ τὸ κινοῦν ὥσπερ εἴρηται πᾶν, τὸ δυνάμει ὂν κινητόν, The mover too is moved, as has been said—every mover, that is, which is capable of motion, καὶ οὗ ἡ ἀκινησία ἠρεμία ἐστίν (ᾧ γὰρ ἡ κίνησις ὑπάρχει, τούτου ἡ ἀκινησία ἠρεμία). and whose immobility is rest—when a thing is subject to motion its immobility is rest. τὸ γὰρ πρὸς τοῦτο ἐνεργεῖν, ᾗ τοιοῦτον, αὐτὸ τὸ κινεῖν ἐστι· τοῦτο δὲ ποιεῖ θίξει, ὥστε ἅμα καὶ πάσχει· For to act on the movable as such is just to move it. But this it does by contact, so that at the same time it is also acted on. διὸ ἡ κίνησις ἐντελέχεια τοῦ κινητοῦ, ᾗ κινητόν, συμβαίνει δὲ τοῦτο θίξει τοῦ κινητικοῦ, ὥσθ' ἅμα καὶ πάσχει. εἶδος δὲ ἀεὶ οἴσεταί τι τὸ κινοῦν, ἤτοι τόδε ἢ τοιόνδε ἢ τοσόνδε, ὃ ἔσται ἀρχὴ καὶ αἴτιον τῆς κινή σεως, ὅταν κινῇ, οἷον ὁ ἐντελεχείᾳ ἄνθρωπος ποιεῖ ἐκ τοῦ δυνάμει ὄντος ἀνθρώπου ἄνθρωπον. Hence we can define motion as the fulfilment of the movable qua movable, the cause of the attribute being contact with what can move so that the mover is also acted on. The mover or agent will always be the vehicle of a form, either a 'this' or 'such', which, when it acts, will be the source and cause of the change, e.g. the full-formed man begets man from what is potentially man. Chapter 3 Καὶ τὸ ἀπορούμενον δὲ φανερόν, ὅτι ἐστὶν ἡ κίνησις ἐν τῷ κινητῷ· ἐντελέχεια γάρ ἐστι τούτου [καὶ] ὑπὸ τοῦ κινητικοῦ. καὶ ἡ τοῦ κινητικοῦ δὲ ἐνέργεια οὐκ ἄλλη ἐστίν· The solution of the difficulty that is raised about the motion—whether it is in the movable—is plain. It is the fulfilment of this potentiality, and by the action of that which has the power of causing motion; and the actuality of that which has the power of causing motion is not other than the actuality of the movable, δεῖ μὲν γὰρ εἶναι ἐντελέχειαν ἀμφοῖν· κινητικὸν μὲν γάρ ἐστιν τῷ δύνασθαι, κινοῦν δὲ τῷ ἐνεργεῖν, for it must be the fulfilment of both. A thing is capable of causing motion because it can do this, it is a mover because it actually does it. ἀλλ' ἔστιν ἐνεργητικὸν τοῦ κινητοῦ, ὥστε ὁμοίως μία ἡ ἀμφοῖν ἐνέργεια But it is on the movable that it is capable of acting. Hence there is a single actuality of both alike, ὥσπερ τὸ αὐτὸ διάστημα ἓν πρὸς δύο καὶ δύο πρὸς ἕν, καὶ τὸ ἄναντες καὶ τὸ κάταντες· ταῦτα γὰρ ἓν μέν ἐστιν, ὁ μέντοι λόγος οὐχ εἷς· ὁμοίως δὲ καὶ ἐπὶ τοῦ κινοῦντος καὶ κινουμένου. just as one to two and two to one are the same interval, and the steep ascent and the steep descent are one—for these are one and the same, although they can be described in different ways. So it is with the mover and the moved.
Postquam philosophus definivit motum, hic ostendit cuius actus sit motus, utrum scilicet mobilis vel moventis. Et potest dici quod hic ponit aliam definitionem motus, quae se habet ad praemissam ut materialis ad formalem, et conclusio ad principium. Et haec est definitio: motus est actus mobilis inquantum est mobile. Haec enim definitio concluditur ex praemissa. Quia enim motus est actus existentis in potentia inquantum huiusmodi; existens autem in potentia inquantum huiusmodi, est mobile, non autem movens, quia movens inquantum huiusmodi est ens in actu; sequitur quod motus sit actus mobilis inquantum huiusmodi. 297. After defining motion, the Philosopher now shows whose act motion is, i.e., whether it is the act of the mobile or of the mover. Also it may be said that he gives another definition of notion which is related to the previous one as material to formal and as a conclusion to its principle. And this is the definition: motion is “the act of the mobile inasmuch as it is mobile.” This definition is a conclusion from the previous one. For since motion is “the act of a thing existing in potency inasmuch as it is in potency,” and since that which exists in potency as such is the mobile and not the mover (for the mover as such is in act), it follows that motion is an act of the mobile as such. Circa hoc ergo tria facit: primo ostendit quod motus est actus mobilis; secundo quomodo se habet motus ad moventem, ibi: motivi autem actus etc.; tertio movet dubitationem, ibi: habet autem defectum et cetera. 298. In regard to the main question he does three things: First he shows that motion is an act of the mobile; Secondly, he shows how motion is related to the mover, at 303; Thirdly, he raises a difficulty, at 308. Circa primum duo facit: primo ponit definitionem motus, scilicet quod motus est actus mobilis; secundo ex hoc declarat quoddam quod poterat esse dubium, ibi: et dubium autem et cetera. About the first he does two things: He posits a definition of motion, namely, that motion is an act of the mobile; He clears up a doubt, at 303. Circa primum tria facit: primo investigat definitionem motus; secundo concludit eam, ibi: unde motus etc.; tertio manifestat eam, ibi: accidit autem et cetera. In regard to (1) he does three things: He investigates the definition of motion: He concludes to the definition, at 302; He explains it, at 302 bis. Ad investigandum autem definitionem motus, praemittit quod moveri etiam accidit moventi. Et circa hoc duo facit: primo probat quod omne movens movetur; secundo ostendit unde accidat ei moveri, ibi: ad hoc enim agere et cetera. In investigating the definition he shows that “to be moved” even occurs to the mover. In regard to this he does two things: He shows that every mover is being moved; He shows why that happens, at 301. Quod autem movens moveatur, ostendit ex duobus. Primo: quidem quia omne quod prius est in potentia et postea in actu, quodammodo movetur; movens autem invenitur prius esse movens in potentia et postea movens in actu; movens ergo huiusmodi movetur. Et hoc est quod dicit, quod omne movens, cum ita se habeat quod quandoque sit in potentia mobile, idest ad movendum, movetur, sicut dictum est: hoc enim ex dictis apparet. Dictum est enim quod motus est actus existentis in potentia; et hoc contingit in omni movente naturali; unde dictum est supra quod omne movens physicum movetur. 299. He shows in two ways that the mover is moved. This is so first of all because anything that is previously in potency and then in act is somehow being moved. But movers are found that are previously movers in potency and afterwards movers in act; therefore they are moved. He states, therefore, that every mover, since it is such that it is in potency to being a mover, is likewise moved. This is clear from what has been already said, for it was said that motion is an act of a thing existing in potency and this occurs in every natural mover; that is why it was said above that every physical mover is moved. Secundo, ibi: et cuius immobilitas etc., ostendit idem alio modo. Cuicumque sua immobilitas est eius quies, huic inest motus; quies enim et motus, cum sint opposita, habent fieri circa idem: sed moventis immobilitas, idest cessatio a movendo, dicitur quies; dicuntur enim quaedam quiescere, quando cessant agere: omne igitur tale movens, scilicet cuius immobilitas est quies, movetur. 300. Secondly  he brings out the same point in another way: Whatsoever’s immobility is its rest is capable of motion; for rest and motion, since they are opposites, happen to the same. But the immobility of a mover, i.e., its ceasing from moving, is called rest; for there are things which said to rest when they cease to act. Therefore every such mover, i.e., one whose immobility is rest, is moved. Deinde cum dicit: ad hoc enim agere etc., ostendit unde accidat moventi quod moveatur. Non enim accidit ei ex hoc quod movet, sed ex hoc quod movet tangendo: quia movere est agere ad hoc quod aliquid moveatur; id autem quod sic a movente patitur, movetur. Sed hoc quod est agere facit tactu; nam corpora tangendo agunt: unde sequitur quod et simul patiatur, quia quod tangit, patitur. Sed hoc intelligendum est quando est mutuus tactus scilicet quod aliquid tangit et tangitur, ut contingit in his quae communicant in materia, quorum utrumque ab altero patitur dum se tangunt. Corpora autem caelestia, quia non communicant cum corporibus inferioribus in materia, sic agunt in ea quod non patiuntur ab eis, et tangunt et non tanguntur, ut dicitur in I de Gen. 301. Then  he shows why it happens that a mover is moved. For it does not happen precisely because it is a mover but because it is such by touching; because to move is to act in order to cause something to be moved and what is so acted upon by the mover is moved. But whatever acts does so by touching, for bodies act by touching; hence it follows that what acts is at the same time acted upon, because that which touches is acted upon. However, this must be understood of those cases where there is mutual touching; namely, when the thing touching is also touched, as happens in things which are material, where both of the things are acted upon when they touch one another. But heavenly bodies, because they do not have material like the lower bodies, so act on them that they are not acted upon in return and they touch without being touched as is stated in De Generatione I (l.18). Deinde cum dicit: unde motus actus mobilis est etc., ponit definitionem motus, concludens ex praedictis quod quamvis movens moveatur, motus tamen non est actus moventis, sed mobilis secundum quod est mobile. Et hoc consequenter manifestat per hoc quod moveri accidit moventi, et non per se ei competit: unde si aliquid secundum hoc movetur, secundum quod actus eius est motus, sequitur quod motus non sit actus moventis, sed mobilis, non quidem inquantum est movens, sed inquantum est mobile. 302. Then  he posits a definition of motion concluding from the aforesaid that although the mover is moved, motion nevertheless is not an act of the mover but of the mobile inasmuch as it is mobile. He shows this subsequently by the fact that “to be moved” is accidental to the mover and does not belong essentially to it. Whence, if something is moved precisely inasmuch as its act is motion it follows that motion is an act not of the mover but of the mobile, not, indeed, insofar as it is a mover but insofar as it is a mobile. Quod autem moveri accidat moventi, manifestat per id quod supra dictum est: hoc enim, scilicet actus mobilis qui est motus, accidit ex contactu moventis: ex quo sequitur quod simul dum agit patiatur, et sic moveri competit moventi per accidens. That “to be moved” is accidental to the mover is clear from what was pointed out in the earlier part of this lecture; for the act of the mobile which is motion happens from its contact with the mover; from which it follows that at the same time that it is acting it is acted upon and thus “to be moved” is accidental to the mover. Quod autem non competat ei per se manifestat per hoc, quod semper aliqua forma videtur movens esse, sicut forma quae est in genere substantiae, in transmutatione quae est secundum substantiam; et forma quae est in genere qualitatis, in alteratione; et forma quae est in genere quantitatis, in augmento et diminutione. Huiusmodi enim formae sunt causae et principia motuum, cum omne agens moveat secundum formam. Omne enim agens agit inquantum est actu, sicut actu homo facit ex homine in potentia hominem actu: unde, cum unumquodque sit actu per formam, sequitur quod forma sit principium movens. Et sic movere competit alicui inquantum habet formam, per quam est in actu. Unde, cum motus sit actus existentis in potentia, ut supra dictum est, sequitur quod motus non sit alicuius inquantum est movens, sed inquantum est mobile: et ideo in definitione motus positum est, quod est actus mobilis inquantum est mobile. That “to be moved” does not belong essentially to the mover is clear from the fact that some form is always seen to be the mover—as the form which is in the genus of substance is the mover in substantial change, and a form in the genus of quality is the mover in alteration, and a form in the genus of quantity in growth and decrease. Forms of this type are the causes and principles of motions, since every agent moves according to its form. For every agent acts inasmuch it is actual—as an actual man makes an actual man of man in potency. Hence, since it is through its form that a thing is actual it follows that form is the moving principle. Thus “to move” belongs to a thing inasmuch as it has a form through which it is actual. Wherefore, since motion. is the act of a thing existing in potency, as said above (l.2, no.285), it follows that motion belongs to a thing not insofar as it is a mover but insofar as it is mobile. For that reason the definition says that motion is an act of the mobile inasmuch as it is mobile. Deinde cum dicit: et dubium autem etc., manifestat quoddam dubium ex praedictis. Solet enim esse dubium apud quosdam, utrum motus sit in movente aut in mobili. Sed hoc dubium declaratur ex praemissis. Manifestum est enim quod actus cuiuslibet est in eo cuius est actus: et sic manifestum est quod actus motus est in mobili, cum sit actus mobilis, causatus tamen in eo a movente. 303. Then  he shows a difficulty that arises from the aforesaid. For some wonder whether motion is in the mover or in the mobile. But this doubt is solved from what went before. For it is clear that an act of anything is in that thing of which it is the act. Thus, it is plain that the act of motion is in the mobile, since it is an act of the mobile, although caused in it by the mover. Deinde cum dicit: motivi autem actus etc., ostendit quomodo se habeat motus ad movens. Et primo proponit quod intendit, dicens quod actus motivi non est alius ab actu mobilis. Unde, cum motus sit actus mobilis, est etiam quodammodo actus motivi. 304. Then [2167 he shows how motion and mover are related. And first of all he proposes his intention, saying that the act of the mover is not distinct from the act of the mobile. Hence since motion is an act of the mobile it is somehow also an act of the mover. Secundo ibi: oportet quidem enim etc., manifestat propositum. Et circa hoc tria facit: primo ostendit quod moventis est aliquis actus, sicut et mobilis. Quidquid enim dicitur secundum potentiam et actum, habet aliquem actum sibi competentem: sed sicut in eo quod movetur dicitur mobile secundum potentiam inquantum potest moveri, motum autem secundum actum inquantum actu movetur; ita ex parte moventis motivum dicitur secundum potentiam, inquantum scilicet potest movere, motus autem in ipso agere, idest in quantum agit actu. Oportet igitur utrique, scilicet moventi et mobili, competere quendam actum. 305. Secondly  he explains this. And in regard to this he does three things. First he shows that there is an act of the mover as well as of the mobile. For whatever is described according to potency and act has some act competent to it. Now just as that which is moved is called “mobile” in potency since it is capable of being moved, and is called “moved” according to act inasmuch it is actually being moved, so on the part of the mover, a mover is described “potential mover” inasmuch as it is able to move, and “moves” in the act inasmuch as it actually acts. Therefore some act is competent to both, i.e., to mover and to mobile. Secundo ibi: sed est activum mobilis etc., ostendit quod idem sit actus moventis et moti. Movens enim dicitur inquantum aliquid agit, motum autem inquantum patitur; sed idem est quod movens agendo causat, et quod motum patiendo recipit. Et hoc est quod dicit, quod movens est activum mobilis, idest actum mobilis causat. Quare oportet unum actum esse utriusque, scilicet moventis et moti: idem enim est quod est a movente ut a causa agente, et quod est in moto ut in patiente et recipiente. 306. Secondly  he shows that the act of the mover and of the mobile is the same act. For something is called “mover” inasmuch as it acts and “moved” inasmuch as it is being acted upon. But what the mover causes by acting and what the moved receives is being acted upon are one and the same. thing. And this is what he means when he says that the mover actualizes the mobile, i.e., causes the act of the mobile. Wherefore, they must both, namely, mover and moved, have the same act; for what is from the mover as agent cause is the same as what is in the moved as patient and receiver. Tertio ibi: sicut idem spatium etc., manifestat hoc per exempla. Eadem enim distantia est unius ad duo et duorum ad unum secundum rem, sed differunt secundum rationem; quia secundum quod incipimus comparationem a duobus procedendo ad unum, dicitur duplum, e contrario vero dicitur dimidium. Et similiter idem est spatium ascendentis et descendentis; sed secundum diversitatem principii et termini, vocatur ascensio vel descensio. Et similiter est in movente, et moto. Nam motus secundum quod procedit a movente in mobile, est actus moventis; secundum autem quod est in mobili a movente, est actus mobilis. 307. Thirdly  he illustrates this by an example. For the distance from one to two is the same as that from two to one, but they differ according to conception; for in relating two to one we have “double,” but in relating one to two we have “one-half.” The same thing is true of the distance covered by one ascending and by one descending, but by reason of the diversity of starting point and term, one is called “ascent” and one “descent.” A parallel case is true of the mover and of the thing moved. For motion, inasmuch as it proceeds from the mover into the mobile, is an act of the mover, but inasmuch as it is in the mobile from the mover, it is an act of the mobile.
Motion as from the agent and in the patient
Chapter 3 cont. ἔχει δ' ἀπορίαν λογικήν· ἀναγκαῖον γὰρ ἴσως εἶναί τινα ἐνέργειαν τοῦ ποιητικοῦ καὶ τοῦ παθητικοῦ· τὸ μὲν δὴ ποίησις, τὸ δὲ πάθησις, ἔργον δὲ καὶ τέλος τοῦ μὲν ποίημα, τοῦ δὲ πάθος. This view has a dialectical difficulty. Perhaps it is necessary that the actuality of the agent and that of the patient should not be the same. The one is 'agency' and the other 'patiency'; and the outcome and completion of the one is an 'action', that of the other a 'passion'. ἐπεὶ οὖν ἄμφω κινήσεις, εἰ μὲν ἕτεραι, ἐν τίνι; ἢ γὰρ ἄμφω ἐν τῷ πάσχοντι καὶ κινουμένῳ, ἢ ἡ μὲν ποίησις ἐν τῷ ποιοῦντι, ἡ δὲ πάθησις ἐν τῷ πάσχοντι (εἰ δὲ δεῖ καὶ ταύτην ποίησιν καλεῖν, ὁμώνυμος ἂν εἴη). Since then they are both motions, we may ask: in what are they, if they are different? Either (a) both are in what is acted on and moved, or (b) the agency is in the agent and the patiency in the patient. (If we ought to call the latter also 'agency', the word would be used in two senses.) ἀλλὰ μὴν εἰ τοῦτο, ἡ κίνησις ἐν τῷ κινοῦντι ἔσται (ὁ γὰρ αὐτὸς λόγος ἐπὶ κινοῦντος καὶ κινουμένου), ὥστ' ἢ πᾶν τὸ κινοῦν κινήσεται, ἢ ἔχον κίνησιν οὐ κινήσεται. Now, in alternative (b), the motion will be in the mover, for the same statement will hold of 'mover' and 'moved'. Hence either every mover will be moved, or, though having motion, it will not be moved. εἰ δ' ἄμφω ἐν τῷ κινουμένῳ καὶ πάσχοντι, καὶ ἡ ποίησις καὶ ἡ πάθησις, καὶ ἡ δίδαξις καὶ ἡ μάθησις δύο οὖσαι ἐν τῷ μανθάνοντι, πρῶτον μὲν ἡ ἐνέργεια ἡ ἑκάστου οὐκ ἐν ἑκάστῳ ὑπάρξει, εἶτα ἄτοπον δύο κινήσεις ἅμα κινεῖσθαι· τίνες γὰρ ἔσονται ἀλλοιώσεις δύο τοῦ ἑνὸς καὶ εἰς ἓν εἶδος; ἀλλ' ἀδύνατον. ἀλλὰ μία ἔσται ἡ ἐνέργεια. If on the other hand (a) both are in what is moved and acted on—both the agency and the patiency (e.g. both teaching and learning, though they are two, in the learner), then, first, the actuality of each will not be present in each, and, a second absurdity, a thing will have two motions at the same time. How will there be two alterations of quality in one subject towards one definite quality? The thing is impossible: the actualization will be one. ἀλλ' (202b.) ἄλογον δύο ἑτέρων τῷ εἴδει τὴν αὐτὴν καὶ μίαν εἶναι ἐνέργειαν· καὶ ἔσται, εἴπερ ἡ δίδαξις καὶ ἡ μάθησις τὸ αὐτὸ καὶ ἡ ποίησις καὶ ἡ πάθησις, καὶ τὸ διδάσκειν τῷ μανθάνειν τὸ αὐτὸ καὶ τὸ ποιεῖν τῷ πάσχειν, ὥστε τὸν διδάσκοντα ἀνάγκη ἔσται πάντα μανθάνειν καὶ τὸν ποιοῦντα πάσχειν. But (some one will say) it is contrary to reason to suppose that there should be one identical actualization of two things which are different in kind. Yet there will be, if teaching and learning are the same, and agency and patiency. To teach will be the same as to learn, and to act the same as to be acted on—the teacher will necessarily be learning everything that he teaches, and the agent will be acted on. One may reply: ἢ οὔτε τὸ τὴν ἄλλου ἐνέργειαν ἐν ἑτέρῳ εἶναι ἄτοπον (ἔστι γὰρ ἡ δίδαξις ἐνέργεια τοῦ διδασκαλικοῦ, ἔν τινι μέντοι, καὶ οὐκ ἀποτετμημένη, ἀλλὰ τοῦδε ἐν τῷδε), (1) It is not absurd that the actualization of one thing should be in another. Teaching is the activity of a person who can teach, yet the operation is performed on some patient—it is not cut adrift from a subject, but is of A on B. οὔτε μίαν δυοῖν κωλύει οὐθὲν τὴν αὐτὴν εἶναι (μὴ ὡς τῷ εἶναι τὸ αὐτό, ἀλλ' ὡς ὑπάρχει τὸ δυνάμει ὂν πρὸς τὸ ἐνεργοῦν), οὔτ' ἀνάγκη τὸν διδά σκοντα μανθάνειν, οὐδ' εἰ τὸ ποιεῖν καὶ πάσχειν τὸ αὐτό ἐστιν, μὴ μέντοι ὥστε τὸν λόγον εἶναι ἕνα τὸν <�τὸ> τί ἦν εἶναι λέγοντα, οἷον ὡς λώπιον καὶ ἱμάτιον, ἀλλ' ὡς ἡ ὁδὸς ἡ Θήβηθεν Ἀθήναζε καὶ ἡ Ἀθήνηθεν εἰς Θήβας, ὥσπερ εἴρηται καὶ πρότερον; οὐ γὰρ ταὐτὰ πάντα ὑπάρχει τοῖς ὁπωσοῦν τοῖς αὐτοῖς, ἀλλὰ μόνον οἷς τὸ εἶναι τὸ αὐτό. (2) There is nothing to prevent two things having one and the same actualization, provided the actualizations are not described in the same way, but are related as what can act to what is acting. (3) Nor is it necessary that the teacher should learn, even if to act and to be acted on are one and the same, provided they are not the same in definition (as 'raiment' and 'dress'), but are the same merely in the sense in which the road from Thebes to Athens and the road from Athens to Thebes are the same, as has been explained above. For it is not things which are in a way the same that have all their attributes the same, but only such as have the same definition. οὐ μὴν ἀλλ' οὐδ' εἰ ἡ δίδαξις τῇ μαθήσει τὸ αὐτό, καὶ τὸ μανθάνειν τῷ διδάσκειν, ὥσπερ οὐδ' εἰ ἡ διάστασις μία τῶν διεστηκότων, καὶ τὸ διίστασθαι ἐνθένθε ἐκεῖσε κἀκεῖθεν δεῦρο ἓν καὶ τὸ αὐτό. But indeed it by no means follows from the fact that teaching is the same as learning, that to learn is the same as to teach, any more than it follows from the fact that there is one distance between two things which are at a distance from each other, that the two vectors AB and Ba, are one and the same. ὅλως δ' εἰπεῖν οὐδ' ἡ δίδαξις τῇ μαθήσει οὐδ' ἡ ποίησις τῇ παθήσει τὸ αὐτὸ κυρίως, ἀλλ' ᾧ ὑπάρχει ταῦτα, ἡ κίνησις· τὸ γὰρ τοῦδε ἐν τῷδε καὶ τὸ τοῦδε ὑπὸ τοῦδε ἐνέργειαν εἶναι ἕτερον τῷ λόγῳ. To generalize, teaching is not the same as learning, or agency as patiency, in the full sense, though they belong to the same subject, the motion; for the 'actualization of X in Y' and the 'actualization of Y through the action of X' differ in definition. τί μὲν οὖν ἐστιν κίνησις εἴρηται καὶ καθόλου καὶ κατὰ μέρος· οὐ γὰρ ἄδηλον πῶς ὁρισθήσεται τῶν εἰδῶν ἕκαστον αὐτῆς· ἀλλοίωσις μὲν γὰρ ἡ τοῦ ἀλλοιωτοῦ, ᾗ ἀλλοιωτόν, ἐντελέχεια. ἔτι δὲ γνωριμώτερον, ἡ τοῦ δυνάμει ποιητικοῦ καὶ παθητικοῦ, ᾗ τοιοῦτον, ἁπλῶς τε καὶ πάλιν καθ' ἕκαστον, ἢ οἰκοδόμησις ἢ ἰάτρευσις. τὸν αὐτὸν δὲ λεχθήσεται τρόπον καὶ περὶ τῶν ἄλλων κινήσεων ἑκάστης. What then Motion is, has been stated both generally and particularly. It is not difficult to see how each of its types will be defined—alteration is the fulfillment of the alterable qua alterable (or, more scientifically, the fulfilment of what can act and what can be acted on, as such)—generally and again in each particular case, building, healing, etc. A similar definition will apply to each of the other kinds of motion.
Postquam philosophus ostendit quod motus est actus mobilis et moventis, nunc movet quandam dubitationem circa praemissa. 308. After showing that motion is the act both of the mobile and of the mover, the Philosopher now raises a difficulty on this point. Et primo movet dubitationem: secundo solvit, ibi: at neque actum et cetera. First, he raises the difficulty: Secondly, he solves it, at 314. Circa primum duo facit: primo praemittit quaedam ad dubitationem; secundo dubitationem prosequitur, ibi: quoniam igitur utraque et cetera. Regarding the first, he does two things: First, he prefaces certain things to the difficulty; Secondly, he builds up the difficulty, at 310. Dicit ergo primo quod id quod dictum est habet defectum, idest dubitationem, rationabilem, idest logicam: ad utramque enim partem sunt probabiles rationes. Ad hanc autem dubitationem hoc praemittit, quod aliquis actus est activi, et aliquis actus est passivi, sicut supra dictum est quod et moventis et moti est aliquis actus. Et actus quidem activi vocatur actio; actus vero passivi vocatur passio. Et hoc probat, quia illud quod est opus et finis uniuscuiusque, est actus eius et perfectio: unde, cum opus et finis agentis sit actio, patientis autem passio, ut per se manifestum est, sequitur quod dictum est, quod actio sit actus agentis et passio patientis. 309. He says therefore  that what has been said above now causes a “rational”, i.e., logical, “defect,” i.e., doubt—by virtue of there being probable reasons for both sides. In introducing the difficulty he says that there is an act in that which is active and there is an act in that which is passive, just as above (no. 305) there was stated to be an act of the mover and of the moved. As a matter of fact, the act of the active is called “action” and the act of the passive is called “passion”. This he proves by saying that the work and end of anything is its act and perfection; hence, since the work and end of the agent is action and that of the patient is passion (or undergoing), it follows that action is the act of the agent and passion that of the patient. Deinde cum dicit: quoniam igitur utraque etc., prosequitur dubitationem. Manifestum est enim quod tam actio quam passio sunt motus: utrumque enim est idem motui. Aut igitur actio et passio sunt idem motus, aut sunt diversi motus. Si sunt diversi, necesse est quod uterque eorum sit in aliquo subiecto: aut igitur uterque est in patiente et moto; aut alter horum est in agente, scilicet actio, et alter in patiente, scilicet passio. Si autem aliquis dicat e converso quod id quod est in agente sit passio, et id quod est in patiente sit actio, manifestum est quod aequivoce loquitur: id enim quod est passio vocabit actionem, et e converso. Videtur autem quartum membrum omittere, scilicet quod utrumque sit in agente: sed hoc praetermittit quia ostensum est quod motus sit in mobili, per quod excluditur hoc membrum, quod neutrum sit in patiente, sed utrumque in agente. 310. Then  he develops this doubt. For it is clear that both action and passion are motion; for each is the same as motion. Therefore, action and passion are either the same motion or diverse motions. If they are diverse, then each of them must be in some subject. Either both will be in the patient, i.e., the thing moved, or one of them (action) is in the agent and the other (passion) is in the patient. To say the opposite, i.e., that what is in the agent is passion and what is in the patient is action is to speak equivocally, or it would be calling passion action and vice versa. The fourth possibility, namely, that both are in the agent is left out, but this is because it has already been shown (nos. 302-303) that motion is in the mobile, which excludes the fourth possibility that neither be in the patient but both in the agent. Ex his igitur duobus membris quae tangit, primo prosequitur secundum, ibi: at vero si hoc est et cetera. Si enim aliquis dicat quod actio est in agente et passio in patiente; actio autem est motus quidam, ut dictum est; sequitur quod motus sit in movente. Eandem autem rationem oportet esse et de movente et de moto, scilicet ut in quocumque eorum sit motus, illud moveatur. Vel eadem ratio est de movente et moto, sicut et de patiente et agente. In quocumque autem est motus, illud movetur; quare sequitur vel quod omne movens moveatur, vel quod aliquid habeat motum et non moveatur; quorum utrumque videtur inconveniens. 311. Of the two possibilities listed, he develops the second one first . For if anyone says that action is in the agent and passion in the patient, then since action is a certain motion, as was stated (no. 310), it follows that motion is in the mover. For the same thing should be true both of the mover and of the moved, namely, that if motion is in either one it is being moved. Or else, that is true of the mover and of the moved which is true of the patient and of the agent. Now, if motion is in something, that thing is being moved; wherefore, it follows that either every mover is being moved or that something has motion but is not being moved; each of these seems unreasonable. Deinde cum dicit: si autem utraque etc., prosequitur aliud membrum, dicens quod si aliquis dicat quod utrumque, scilicet actio et passio, cum sint duo motus, sunt in patiente et moto; et doctio, quae est ex parte docentis, et doctrina, quae est ex parte addiscentis, sunt in addiscente; sequuntur duo inconvenientia. Quorum primum est quia dictum est quod actio est actus agentis: si igitur actio non est in agente sed in patiente, sequetur quod proprius actus uniuscuiusque non est in eo cuius est actus. 312. Then  he develops the second possibility given in 310. He says that if anyone should say that both of them, namely, action and passion, since they are two motions, are in the patient, which is equivalent to saying that teaching which is on the part of the teacher and learning which is on the part of the learner are both in the learner, then two conflicts arise. The first is that if what we said previously is true, namely, that action is an act of the agent, then if action is not in the agent but in the patient, it will follow that the proper act of each thing is not in the thing of which it is the act. Postea sequetur aliud inconveniens, scilicet quod aliquid unum et idem moveatur secundum duos motus. Actio enim et passio supponuntur nunc esse duo motus; in quocumque autem est motus, illud movetur secundum illum motum; si igitur actio et passio sunt in mobili, sequitur quod mobile moveatur secundum duos motus. Et hoc idem esset ac si essent duae alterationes unius subiecti, quae terminarentur ad unam speciem, sicut quod unum subiectum moveretur duabus dealbationibus; quod est impossibile. Quod vero idem subiectum moveatur duabus alterationibus simul, ad diversas species terminatis, scilicet dealbatione et calefactione, non est inconveniens. Manifestum autem est quod actio et passio ad eandem speciem terminantur: idem est enim quod agens agit et patiens patitur. Then another conflict follows, namely, that one and the same thing is being moved according to two motions. For action and passion are now supposed to be two motions. Now in whatever thing there is a motion that thing is being moved according to that motion; if then action and passion are in the mobile, it follows that the mobile is being moved according to two motions. This would be tantamount to having two alterations in one subject both of them specifically the same; for example, one subject being moved to two whitenings, which is impossible. This does not mean that one subject could not be moved by two alterations tending toward two specifically different terms, for example, whitening and heating. Nevertheless, it is clear that action and passion terminate at the same specific term; for what the agent does and what the patient receives are one and the same. Deinde cum dicit: sed unus erit actus? et cetera, prosequitur aliud membrum. Potest enim aliquis dicere quod actio et passio non sunt duo motus, sed unus. Et ex hoc ducit ad quatuor inconvenientia. Quorum primum est, quod idem sit actus diversorum secundum speciem. Dictum est enim quod actio sit actus agentis, et passio actus patientis, quae secundum speciem sunt diversa: si igitur actio et passio sint idem motus, sequitur quod idem actus sit diversorum secundum speciem. Secundum inconveniens est, quod si actio et passio sint unus motus, quod idem sit actio cum passione, et doctio, quae est ex parte docentis, cum doctrina secundum quod se tenet ex parte addiscentis. Tertium inconveniens est, quod agere sit idem quod pati, et docere idem quod addiscere. Quartum quod ex hoc sequitur, est quod omne docens addiscat, et omne agens patiatur. 313. Then  he develops the other possibility. For it could be said that action and passion are not two motions but one. But this leads to four difficulties. The first is that the act of things of different species would be the same. For it has been already pointed out (no. 309) that action is an act of the agent and passion ant act of the patient and that these are specifically diverse; but if action and passion are the same motion then the act of specifically different things will be the same. The second difficulty is that if action and passion are one motion, then action is the same as passion, so that teaching which is laid to the teacher is the same as learning which is in the learner. The third difficulty is that acting would be the same as being acted upon and teaching would be the same as learning. The fourth difficulty that follows from this is that every teacher would be learning and every agent would be being acted upon. Deinde cum dicit: at neque actum alterius etc., solvit praemissam dubitationem. Est autem manifestum ex supra determinatis quod actio et passio non sunt duo motus, sed unus et idem motus: secundum enim quod est ab agente dicitur actio, secundum autem quod est in patiente dicitur passio. 314. Then  he solves the difficulty. From what was settled previously, (nos. 304,306) it is clear that action and passion are not two motions but one and the same motion; for insofar as motion is from the agent it is called “action,” and insofar as it is in the patient it is called “passion.” Unde inconvenientia quae sequuntur ad primam partem, in qua supponebatur quod actio et passio essent duo motus, non oportet solvere, praeter unum, quod remanet solvendum, etiam supposito quod actio et passio sint unus motus: quia cum actio sit actus agentis, ut supra dictum est, si actio et passio sunt unus motus, sequitur quod actus agentis quodammodo sit in patiente, et sic actus unius erit in altero. Quatuor autem inconvenientia sequebantur ex altera parte; et sic restant quinque inconvenientia solvenda. 315. Hence not all the conflicts which follow from the first case, in which it was supposed that action and passion are two motions, have to be solved. But one remains to be solved even on the supposition that action and passion are one motion: because since action is an act of the agent, then if action and passion are one motion, it follows that the act of the agent is somehow in the patient and thus the act of one thing will be in something else. This remaining difficulty together with the four listed in 313 leave five to be solved. Dicit ergo primo quod non est inconveniens actum unius esse in altero, quia doctio est actus docentis, ab eo tamen in alterum tendens continue et sine aliqua interruptione: unde idem actus est huius, idest agentis, ut a quo; et tamen est in patiente ut receptus in eo. Esset autem inconveniens si actus unius eo modo quo est actus eius, esset in alio. 316. He says in the first place that there is nothing wrong with an act of one thing being in something else, for teaching is an act of the teacher, an act continuously tending from him into someone else without interruption; hence, this act which is the agent’s as being “from which” is the very one which is in the patient as received in him. But it would be wrong if the act of the one were the act of the other in precisely the same way. Deinde cum dicit: neque unum duobus etc., solvit aliud inconveniens, scilicet quod idem actus esset duorum. Et dicit quod nihil prohibet unum actum esse duorum, ita quod non sit unus et idem secundum rationem, sed unus secundum rem, ut dictum est supra quod eadem est distantia duorum ad unum et unius ad duo, et eius quod est in potentia ad agens et e converso. Sic enim idem actus secundum rem est duorum secundum diversam rationem: agentis quidem secundum quod est ab eo, patientis autem secundum quod est in ipso. 317. Then  he solves another difficulty; namely, that there would be one and the same act for two diverse things. And he says that there is nothing to prevent one act belonging to two things so long as it is not one and the same in aspect but only in reality, as was already explained above (no. 307) when it was pointed out that the distance from one to two and from two to one are the same; and of that which is in potency looking toward the agent and conversely. For in these cases the same one reality is assigned to two things according to different aspects: it is assigned to the agent inasmuch as it is from it and to the patient inasmuch as it is in it. Ad alia autem tria inconvenientia, quorum unum ex altero deducebatur, respondet ordine retrogrado. Primo scilicet ad illud quod ultimo inducebatur, ut magis inconveniens. Sic igitur tertio respondet ad quintum inconveniens. Et dicit quod non est necessarium quod docens addiscat, vel quod agens patiatur, etsi agere et pati sint idem; dum tamen dicamus quod non sunt idem sicut ea quorum ratio est una, ut tunica et indumentum, sed sicut ea quae sunt idem subiecto et diversa secundum rationem, ut via a Thebis ad Athenas et ab Athenis ad Thebas, ut dictum est prius. Non enim oportet quod omnia eadem conveniant iis quae sunt quocumque modo idem; sed solum illis quae sunt idem subiecto vel re et ratione. Et ideo, etiam dato quod agere et pati sint idem, cum non sint idem ratione, ut dictum est, non sequitur quod cuicumque convenit agere, quod ei conveniat pati. 318. The three remaining difficulties of which one followed logically from the other he takes care of in reverse order. He disposes first of the last difficulty deduced, because it is so evidently improper. Thus he is now, thirdly, settling the fifth difficulty. He says that it is not necessary to say that one who is teaching is learning or that an agent is being acted upon just because to act and to be acted upon are the same, as long as we understand that they are not the same in the way that dress and clothing are the same (for these are the same in motion) but in the way, as said above (nos. 307,318), that the road from Thebes to Athens and from Athens to Thebes are the same, i.e., as being the same as to subject but differing as to notion. For it is not necessary that things which are somehow the same should be the same in all ways; that is true only of things that are the same in subject or reality and also in motion. And therefore even granting that to act and to be acted upon are the same, yet since they are not the same in notion, it will not follow that it is the same for an object to act and to be acted upon. Deinde cum dicit: at vero neque si doctio etc., respondet ad quartum inconveniens. Et dicit quod non sequitur, etiam si doctio et doctrina addiscentis essent idem, quod docere et addiscere essent idem; quia doctio et doctrina dicuntur in abstracto, docere autem et discere in concreto. Unde applicantur ad fines vel ad terminos, secundum quos sumitur diversa ratio actionis et passionis. Sicut licet dicamus quod sit idem spatium distantium aliquorum, abstracte accipiendo; si tamen applicemus ad terminos spatii, sicut cum dicimus distare hinc illuc et inde huc, non est unum et idem. 319. Then  he answers the fourth difficulty. And he says that even though teaching and the doctrine of the learner were the same, it does not follow that to teach and to learn are the same; because teaching and doctrine are abstract terms, whereas to teach and to learn are concrete. Hence they are being applied to ends or to terms which serve as the basis for the difference in notion between action and passion. For just as the distance between two points is one and the same space in the abstract, yet if we apply it to two concrete places it is not one and the same, as when we say that there is a distance between here and there and between there and here. Deinde cum dicit: omnino autem dicere est etc., respondet ad tertium inconveniens destruens hanc illationem, qua concludebatur quod si actio et passio sunt unus motus, quod actio et passio sunt idem. Et dicit quod finaliter dicendum est, quod non sequitur quod actio et passio sint idem, vel doctio et doctrina, sed quod motus cui inest utrumque eorum, sit idem. Qui quidem motus secundum unam rationem est actio, et secundum aliam rationem est passio. Alterum enim est secundum rationem esse actum huius ut in hoc, et esse actum huius ut ab hoc. Motus autem dicitur actio secundum quod est actus agentis ut ab hoc: dicitur autem passio secundum quod est actus patientis ut in hoc. Et sic patet quod licet motus sit idem moventis et moti, propter hoc quod abstrahit ab utraque ratione, tamen actio et passio differunt propter hoc, quod has diversas rationes in sua significatione includunt. Ex hoc autem apparet quod, cum motus abstrahat a ratione actionis et passionis, non continetur in praedicamento actionis neque in praedicamento passionis, ut quidam dixerunt. 320. Then  he answers the third difficulty by destroying the inference that if action and passion are one motion, they are the same. And he says it necessary to say finally that it does not follow that action and passion are the same or that teaching and learning are, but rather that the motion in which both are is the same. This motion as a matter of fact is action from one viewpoint and passion from another. For it is one thing as to notion to be an act of a thing as being in it and another to be the act of a thing as being from it. Now motion is called “action” inasmuch as it is an act of the agent as from the agent; it is called “passion” inasmuch as it is an act of the patient as in the patient. Thus it is clear that although the motion of the mover and of the moved is the same thing due to the fact that motion as such abstracts from these aspects, yet action and passion differ due to the fact that these aspects are included in their signification. From this it is also apparent that since motion abstracts from the notion of action and passion, it belongs neither in the predicament “action” nor in the predicament “passion,” as some supposed. Sed restat circa hoc duplex dubitatio. Prima quidem quia, si actio et passio sint unus motus, et non differunt nisi secundum rationem, ut dictum est, videtur quod non debeant esse duo praedicamenta, cum praedicamenta sint genera rerum. Item, si motus vel est actio vel passio, non invenietur motus in substantia, qualitate, quantitate et ubi, ut supra dictum est; sed solum continebitur in actione et passione. 321. But two difficulties still remain with respect to this. The first is this: if action and passion are one motion, and they differ merely in thought, as said above (no. 317), it seems that they should not be listed as two distinct predicaments, since the predicaments are genera of things. Secondly, if motion is either action or passion, motion will not be found in substance, quality, quantity, and place, as said above (no. 286), but only in action and passion. Ad horum igitur evidentiam sciendum est quod ens dividitur in decem praedicamenta non univoce, sicut genus in species, sed secundum diversum modum essendi. Modi autem essendi proportionales sunt modis praedicandi. Praedicando enim aliquid de aliquo altero, dicimus hoc esse illud: unde et decem genera entis dicuntur decem praedicamenta. 322. To settle this matter it must be remembered that being is divided into the ten predicaments not univocally, as a genus into its species, but according to the diverse manner of existing. Now the modes of existing are parallel to the modes of predicating. For in predicating something of something, we say that this is that; that is why the ten genera of being are called “predicaments.” Tripliciter autem fit omnis praedicatio. Unus quidem modus est, quando de aliquo subiecto praedicatur id quod pertinet ad essentiam eius, ut cum dico Socrates est homo, vel homo est animal; et secundum hoc accipitur praedicamentum substantiae. Now every predication takes place in one of three ways. One way is to predicate of a subject that which pertains to its essence, as when I say “Socrates is man” or “Man is animal.” According to this the predicament of “substance” is taken. Alius autem modus est quo praedicatur de aliquo id quod non est de essentia eius, tamen inhaeret ei. Quod quidem vel se habet ex parte materiae subiecti, et secundum hoc est praedicamentum quantitatis (nam quantitas proprie consequitur materiam: unde et Plato posuit magnum ex parte materiae); aut consequitur formam, et sic est praedicamentum qualitatis (unde et qualitates fundantur super quantitatem, sicut color in superficie, et figura in lineis vel in superficiebus); aut se habet per respectum ad alterum, et sic est praedicamentum relationis (cum enim dico homo est pater, non praedicatur de homine aliquid absolutum, sed respectus qui ei inest ad aliquid extrinsecum). Another way is to predicate of a subject something that is not of its essence but yet inheres in the subject, This inherent thing may be traceable to the matter in the subject, in which case one has the predicament of “quantity” (for quantity is properly a result of matter; for which reason Plato traced the “large” to matter); or it is traceable to the form and in this case, there is the predicament of “quality” (for which reason qualities are founded on quality, as color in a surface, and figure in lines or in a plane); or the predication may be due to a relation existing between subject and something else and thus we have the predicament of “relation”, (for when I say, “The man is a father,” it is not something absolute that is predicated of the man but a relation in him to something without). Tertius autem modus praedicandi est, quando aliquid extrinsecum de aliquo praedicatur per modum alicuius denominationis: sic enim et accidentia extrinseca de substantiis praedicantur; non tamen dicimus quod homo sit albedo, sed quod homo sit albus. Denominari autem ab aliquo extrinseco invenitur quidem quodammodo communiter in omnibus, et aliquo modo specialiter in iis quae ad homines pertinent tantum. The third mode of predicating is when something outside the subject is predicated after the manner of denomination; this allows even extrinsic accidents to be predicated of substance; but yet we do not say that man is whiteness but that man is white. To be denominated by something extrinsic can occur, generally speaking, to all things in one way or another, and in a special way in those matters that refer only to man. Communiter autem invenitur aliquid denominari ab aliquo extrinseco, vel secundum rationem causae, vel secundum rationem mensurae; denominatur enim aliquid causatum et mensuratum ab aliquo exteriori. Cum autem quatuor sint genera causarum, duo ex his sunt partes essentiae, scilicet materia et forma: unde praedicatio quae posset fieri secundum haec duo, pertinet ad praedicamentum substantiae, utpote si dicamus quod homo est rationalis, et homo est corporeus. Causa autem finalis non causat seorsum aliquid ab agente: intantum enim finis habet rationem causae, inquantum movet agentem. Remanet igitur sola causa agens a qua potest denominari aliquid sicut ab exteriori. Sic igitur secundum quod aliquid denominatur a causa agente, est praedicamentum passionis, nam pati nihil est aliud quam suscipere aliquid ab agente: secundum autem quod e converso denominatur causa agens ab effectu, est praedicamentum actionis, nam actio est actus ab agente in aliud, ut supra dictum est. Speaking generally, a thing can be denominated by something extrinsic either according to the notion of cause or according to that of measure. For something is denominated “caused” or “measured” on account of its relationship to something extrinsic. Now there are four genera of causes, two of which are parts of the essence, namely, matter and form; hence any predication based on these two pertains to the predicament of “substance,” as when I say that man is rational and man is corporeal. In regard to the other two causes, the final cause does not cause separately from the agent; for the end is a cause only insofar as it influences the agent. Therefore, the only cause according to which a thing can be denominated something as based on something extrinsic is the agent cause. Consequently, when something is denominated from the agent cause, it is the predicament of “passion,” for to undergo (pati) is nothing but the undergoing of something from an agent; on the other hand, if the agent cause is denominated something on account of its effect, one has the predicament of “action,” for action is an act from the agent into something else, as stated above (no, 316). Mensura autem quaedam est extrinseca et quaedam intrinseca. Intrinseca quidem sicut propria longitudo uniuscuiusque et latitudo et profunditas: ab his ergo denominatur aliquid sicut ab intrinseco inhaerente; unde pertinet ad praedicamentum quantitatis. Exteriores autem mensurae sunt tempus et locus: secundum igitur quod aliquid denominatur a tempore, est praedicamentum quando; secundum autem quod denominatur a loco, est praedicamentum ubi et situs, quod addit supra ubi ordinem partium in loco. Hoc autem non erat necessarium addi ex parte temporis, cum ordo partium in tempore in ratione temporis importetur: est enim tempus numerus motus secundum prius et posterius. Sic igitur aliquid dicitur esse quando vel ubi per denominationem a tempore vel a loco. In regard to measures, it will be either intrinsic or extrinsic. An intrinsic measure would be a thing’s own length and width and depth: in these cases a subject is being denominated something by reason of what inheres intrinsically; hence this Pertains to the predicament quantity. The extrinsic measures are time and place. It is the predicament “when”, whenever something is denominated by time; when it is denominated by place, it is the predicament “where” or the predicament “situs”, which adds to “where” the order of the parts in place. Such an order of parts is not considered in regard to the measure which is time, for the order of parts in time in time is already implied in the notion of time; for time is the number of motion according to the order of the “before” and the “after” [its parts]. Thus it is through denomination from time or place that something is said to be “when” or “where”. Est autem aliquid speciale in hominibus. In aliis enim animalibus natura dedit sufficienter ea quae ad conservationem vitae pertinent, ut cornua ad defendendum, corium grossum et pilosum ad tegendum, ungulas vel aliquid huiusmodi ad incedendum sine laesione. Et sic cum talia animalia dicuntur armata vel vestita vel calceata, quodammodo non denominantur ab aliquo extrinseco, sed ab aliquibus suis partibus. Unde hoc refertur in his ad praedicamentum substantiae: ut puta si diceretur quod homo est manuatus vel pedatus. Sed huiusmodi non poterant dari homini a natura, tum quia non conveniebant subtilitati complexionis eius, tum propter multiformitatem operum quae conveniunt homini inquantum habet rationem, quibus aliqua determinata instrumenta accommodari non poterant a natura: sed loco omnium inest homini ratio, qua exteriora sibi praeparat loco horum quae aliis animalibus intrinseca sunt. Unde cum homo dicitur armatus vel vestitus vel calceatus, denominatur ab aliquo extrinseco, quod non habet rationem neque causae, neque mensurae: unde est speciale praedicamentum, et dicitur habitus. Sed attendendum est quod etiam aliis animalibus hoc praedicamentum attribuitur, non secundum quod in sua natura considerantur, sed secundum quod in hominis usum veniunt; ut si dicamus equum phaleratum vel sellatum seu armatum. There is a special predicament for men. For in other animals nature provided the requirements for preserving life, such as horns for defense, a tough and wooly hide as a covering, claws or the like for proceeding without harm. Hence, when by reason of this equipment animals are said to be “armed” or “covered” or “shod,” they are somehow so called not by reason of something; extrinsic but of something intrinsic, which is part of them. Hence, such are referred to the predicament of “substance,” as the same would be if man were said to be “endowed with hands” or “feet.” But the other things could not be endowed upon man by nature, both because they would be out of keeping with the subtlety of his complexion and because reason makes man capable of an enormous number of works for the performance of which nature could not have endowed him with specific instruments. In the place of all these instruments man has reason, which he can use to make for himself the things that are intrinsic to other animals. So when a man is said to be armed or clothed or shod, he is denominated thus by reason of something extrinsic to him that is neither a cause nor a measure; hence it is located in a special predicament called “habitus.” But we should not fail to note that this predicament is in certain matters used also for other animals not inasmuch as they are considered in their nature but insofar as they are put at the service of man: thus we that a horse is caparisoned or saddled or armed. Sic igitur patet quod licet motus sit unus, tamen praedicamenta quae sumuntur secundum motum, sunt duo, secundum quod a diversis rebus exterioribus fiunt praedicamentales denominationes. Nam alia res est agens, a qua sicut ab exteriori, sumitur per modum denominationis praedicamentum passionis: et alia res est patiens a qua denominatur agens. Et sic patet solutio primae dubitationis. 323. This makes it clear that although motion is one, yet there are two predicaments which are based on motion depending on the different external things according to which the predicamental denominations are made. For an agent is one thing from which as from something external the predicament of “passion” is taken; and the patient is some other thing from which something in denominated an agent. This solves the first difficulty (mentioned in 321). Secunda autem dubitatio de facili solvitur. Nam ratio motus completur non solum per id quod est de motu in rerum natura, sed etiam per id quod ratio apprehendit. De motu enim in rerum natura nihil aliud est quam actus imperfectus, qui est inchoatio quaedam actus perfecti in eo quod movetur: sicut in eo quod dealbatur, iam incipit esse aliquid albedinis. Sed ad hoc quod illud imperfectum habeat rationem motus, requiritur ulterius quod intelligamus ipsum quasi medium inter duo; quorum praecedens comparatur ad ipsum sicut potentia ad actum, unde motus dicitur actus; consequens vero comparatur ad ipsum sicut perfectum ad imperfectum vel actus ad potentiam, propter quod dicitur actus existentis in potentia, ut supra dictum est. Unde quodcumque imperfectum accipiatur ut non in aliud perfectum tendens, dicitur terminus motus et non erit motus secundum quem aliquid moveatur; utpote si aliquid incipiat dealbari, et statim alteratio interrumpatur. 324. The second doubt is easy to solve. For the idea of motion depends not only on that which pertains to motion in reality but also on that which reason apprehends. In reality, motion is nothing more than an imperfect act which is a sort of beginning of a perfect act in that which is being moved; thus, in that which is becoming white, some whiteness has begun to be. But in order that what is imperfect have the aspect of motion it is further required that we understand it as a medium between two: the preceding one of them is compared to motion as potency to act (whence motion is called act); the consequent one is compared to motion as the perfect to the imperfect or as act to potency, wherefore motion is called “the act of a being that exists in potency,” as we said above (no. 285). But anything imperfect, if it is not considered to be tending on to something other as perfect, is called the terminus of motion and one will not have a motion according to which something is being moved; as, for example, if something should start to become white and then the alteration was immediately stopped. Quantum igitur ad id quod in rerum natura est de motu, motus ponitur per reductionem in illo genere quod terminat motum, sicut imperfectum reducitur ad perfectum, ut supra dictum est. Sed quantum ad id quod ratio apprehendit circa motum, scilicet esse medium quoddam inter duos terminos, sic iam implicatur ratio causae et effectus: nam reduci aliquid de potentia in actum, non est nisi ab aliqua causa agente. Et secundum hoc motus pertinet ad praedicamentum actionis et passionis: haec enim duo praedicamenta accipiuntur secundum rationem causae agentis et effectus, ut dictum est. Therefore, in regard to what there is of motion in external reality, motion is placed reductively in that genus which terminates the motion, as the imperfect is reduced to the perfect, as stated above (no. 281). But in regard to what reason apprehends about motion, namely, that it is midway between two-terms, here the notion of cause and effect are brought in; because for something to be reduced from potency to act an agent cause is required. From this aspect, motion pertains to the predicaments of “action” and “passion”; for these two predicaments are based on the notions of acting cause and of effect, as was said above (no. 322). Deinde cum dicit: quid quidem igitur motus etc., definit motum in particulari: et dicit quod dictum est quid sit motus et in universali et in particulari; quia ex hoc quod dictum est de definitione motus in universali, manifestum esse poterit quomodo definiatur in particulari. Si enim motus est actus mobilis secundum quod huiusmodi, sequitur quod alteratio sit actus alterabilis secundum quod huiusmodi: et sic de aliis. 325. Then  he defines motion more particularly. He says that we have pointed out what motion is both in general and in particular—because from what was said about the definition of motion in general is clear how it can be defined in particular. For if motion is the act of the mobile as such, it follows that alteration is the act of the, alterable as alterable, and so on for other particular kinds of motion. Et quia positum fuit in dubitatione, utrum motus sit actus moventis vel mobilis, et ostensum est quod est actus activi ut ab hoc, et passivi ut in hoc; ad tollendum omnem dubitationem aliquantulum notius dicamus quod motus est actus potentiae activi et passivi. And because there was a doubt whether motion is an act of the mover or of the mobile and we showed(no. 320) that it is an act of the active as from it and of the passive as in it, then to remove any further doubts we can say somewhat more explicitly that motion is an act of the potency of that which is active and of that which is passive. Et sic etiam poterimus in particulari dicere quod aedificatio est actus aedificatoris et aedificabilis inquantum huiusmodi: et simile est de medicatione et aliis motibus. In this way we could have said that building is an act of the “builder” and. of the “buildable as buildable”; the same is true of healing and of other motions.
Early opinions on the infinite
Chapter 4 Ἐπεὶ δ' ἐστὶν ἡ περὶ φύσεως ἐπιστήμη περὶ μεγέθη καὶ κίνησιν καὶ χρόνον, ὧν ἕκαστον ἀναγκαῖον ἢ ἄπειρον ἢ πεπερασμένον εἶναι, εἰ καὶ μὴ πᾶν ἐστιν ἄπειρον ἢ πεπερασμένον, οἷον πάθος ἢ στιγμή (τῶν γὰρ τοιούτων ἴσως οὐδὲν ἀναγκαῖον ἐν θατέρῳ τούτων εἶναι), προσῆκον ἂν εἴη τὸν περὶ φύσεως πραγματευόμενον θεωρῆσαι περὶ ἀπείρου, εἰ ἔστιν ἢ μή, καὶ εἰ ἔστιν, τί ἐστιν. The science of nature is concerned with spatial magnitudes and motion and time, and each of these at least is necessarily infinite or finite, even if some things dealt with by the science are not, e.g. a quality or a point—it is not necessary perhaps that such things should be put under either head. Hence it is incumbent on the person who specializes in physics to discuss the infinite and to inquire whether there is such a thing or not, and, if there is, what it is. σημεῖον δ' ὅτι ταύτης τῆς (203a.) ἐπιστήμης οἰκεία ἡ θεωρία ἡ περὶ αὐτοῦ· πάντες γὰρ οἱ δοκοῦντες ἀξιολόγως ἧφθαι τῆς τοιαύτης φιλοσοφίας πεποίηνται λόγον περὶ τοῦ ἀπείρου, καὶ πάντες ὡς ἀρχήν τινα τιθέασι τῶν ὄντων, The appropriateness to the science of this problem is clearly indicated. All who have touched on this kind of science in a way worth considering have formulated views about the infinite, and indeed, to a man, make it a principle of things. οἱ μέν, ὥσπερ οἱ Πυθαγόρειοι καὶ Πλάτων, καθ' αὑτό, οὐχ ὡς συμβεβηκός τινι ἑτέρῳ ἀλλ' οὐσίαν αὐτὸ ὂν τὸ ἄπει ρον. (1) Some, as the Pythagoreans and Plato, make the infinite a principle in the sense of a self-subsistent substance, and not as a mere attribute of some other thing. πλὴν οἱ μὲν Πυθαγόρειοι ἐν τοῖς αἰσθητοῖς (οὐ γὰρ χωριστὸν ποιοῦσιν τὸν ἀριθμόν), καὶ εἶναι τὸ ἔξω τοῦ οὐρανοῦ ἄπειρον, Πλάτων δὲ ἔξω μὲν οὐδὲν εἶναι σῶμα, οὐδὲ τὰς ἰδέας, διὰ τὸ μηδὲ ποὺ εἶναι αὐτάς, τὸ μέντοι ἄπειρον καὶ ἐν τοῖς αἰσθητοῖς καὶ ἐν ἐκείναις εἶναι· Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also. καὶ οἱ μὲν τὸ ἄπειρον εἶναι τὸ ἄρτιον (τοῦτο γὰρ ἐναπολαμβανόμενον καὶ ὑπὸ τοῦ περιττοῦ περαινόμενον παρέχειν τοῖς οὖσι τὴν ἀπειρίαν· σημεῖον δ' εἶναι τούτου τὸ συμβαῖνον ἐπὶ τῶν ἀριθμῶν· περιτιθεμένων γὰρ τῶν γνωμόνων περὶ τὸ ἓν καὶ χωρὶς ὁτὲ μὲν ἄλλο ἀεὶ γίγνεσθαι τὸ εἶδος, ὁτὲ δὲ ἕν), Πλάτων δὲ δύο τὰ ἄπειρα, τὸ μέγα καὶ τὸ μικρόν. Further, the Pythagoreans identify the infinite with the even. For this, they say, when it is cut off and shut in by the odd, provides things with the element of infinity. An indication of this is what happens with numbers. If the gnomons are placed round the one, and without the one, in the one construction the figure that results is always different, in the other it is always the same. But Plato has two infinites, the Great and the Small. οἱ δὲ περὶ φύσεως πάντες [ἀεὶ] ὑποτιθέασιν ἑτέραν τινὰ φύσιν τῷ ἀπείρῳ τῶν λεγομένων στοιχείων, οἷον ὕδωρ ἢ ἀέρα ἢ τὸ μεταξὺ τούτων. τῶν δὲ πεπερασμένα ποιούντων στοιχεῖα οὐθεὶς ἄπειρα ποιεῖ· ὅσοι δ' ἄπειρα ποιοῦσι τὰ στοιχεῖα, καθάπερ Ἀναξαγόρας καὶ Δημόκριτος, ὁ μὲν ἐκ τῶν ὁμοιομερῶν, ὁ δ' ἐκ τῆς πανσπερμίας τῶν σχημάτων, τῇ ἁφῇ συνεχὲς τὸ ἄπειρον εἶναι φασίν· καὶ ὁ μὲν ὁτιοῦν τῶν μορίων εἶναι μίγμα ὁμοίως τῷ παντὶ διὰ τὸ ὁρᾶν ὁτιοῦν ἐξ ὁτουοῦν γιγνόμενον· ἐντεῦθεν γὰρ ἔοικε καὶ ὁμοῦ ποτὲ πάντα χρήματα φάναι εἶναι, οἷον ἥδε ἡ σὰρξ καὶ τόδε τὸ ὁστοῦν, καὶ οὕτως ὁτιοῦν· καὶ πάντα ἄρα· καὶ ἅμα τοίνυν· ἀρχὴ γὰρ οὐ μόνον ἐν ἑκάστῳ ἔστι τῆς διακρίσεως, ἀλλὰ καὶ πάντων. ἐπεὶ γὰρ τὸ γιγνόμενον ἐκ τοῦ τοιούτου γίγνεται σώματος, πάντων δ' ἔστι γένεσις πλὴν οὐχ ἅμα, καί τινα ἀρχὴν δεῖ εἶναι τῆς γενέσεως, αὕτη δ' ἐστὶν μία, οἷον ἐκεῖνος καλεῖ νοῦν, ὁ δὲ νοῦς ἀπ' ἀρχῆς τινος ἐργάζεται νοήσας· ὥστε ἀνάγκη ὁμοῦ ποτε πάντα εἶναι καὶ ἄρξασθαί ποτε κινούμενα. #916;ημόκριτος δ' οὐδὲν ἕτερον ἐξ ἑτέρου γίγνεσθαι τῶν πρώτων φησίν· ἀλλ' ὅμως γε αὐτῷ τὸ κοινὸν (203b.) σῶμα πάντων ἐστὶν ἀρχή. μεγέθει κατὰ μόρια καὶ σχήματι διαφέρον. The physicists, on the other hand, all of them, always regard the infinite as an attribute of a substance which is different from it and belongs to the class of the so-called elements—water or air or what is intermediate between them. Those who make them limited in number never make them infinite in amount. But those who make the elements infinite in number, as Anaxagoras and Democritus do, say that the infinite is continuous by contact—compounded of the homogeneous parts according to the one, of the seed-mass of the atomic shapes according to the other. Further, Anaxagoras held that any part is a mixture in the same way as the All, on the ground of the observed fact that anything comes out of anything. For it is probably for this reason that he maintains that once upon a time all things were together. (This flesh and this bone were together, and so of any thing: therefore all things: and at the same time too.) For there is a beginning of separation, not only for each thing, but for all. Each thing that comes to be comes from a similar body, and there is a coming to be of all things, though not, it is true, at the same time. Hence there must also be an origin of coming to be. One such source there is which he calls Mind, and Mind begins its work of thinking from some starting-point. So necessarily all things must have been together at a certain time, and must have begun to be moved at a certain time. Democritus, for his part, asserts the contrary, namely that no element arises from another element. Nevertheless for him the common body is a source of all things, differing from part to part in size and in shape. It is clear then from these considerations that the inquiry concerns the physicist. ὅτι μὲν οὖν προσήκουσα τοῖς φυσικοῖς ἡ θεωρία, δῆλον ἐκ τούτων. εὐλόγως δὲ καὶ ἀρχὴν αὐτὸ τιθέασι πάντες· οὔτε γὰρ μάτην οἷόν τε αὐτὸ εἶναι, οὔτε ἄλλην ὑπάρχειν αὐτῷ δύναμιν πλὴν ὡς ἀρχήν· ἅπαντα γὰρ ἢ ἀρχὴ ἢ ἐξ ἀρχῆς, τοῦ δὲ ἀπείρου οὐκ ἔστιν ἀρχή· εἴη γὰρ ἂν αὐτοῦ πέρας. ἔτι δὲ καὶ ἀγένητον καὶ ἄφθαρτον ὡς ἀρχή τις οὖσα· τό τε γὰρ γενόμενον ἀνάγκη τέλος λαβεῖν, καὶ τελευτὴ πάσης ἔστιν φθορᾶς. διό, καθάπερ λέγομεν, οὐ ταύτης ἀρχή, ἀλλ' αὕτη τῶν ἄλλων εἶναι δοκεῖ καὶ περιέχειν ἅπαντα καὶ πάντα κυβερνᾶν, ὥς φασιν ὅσοι μὴ ποιοῦσι παρὰ τὸ ἄπειρον ἄλλας αἰτίας, οἷον νοῦν ἢ φιλίαν· καὶ τοῦτ' εἶναι τὸ θεῖον· ἀθάνατον γὰρ καὶ ἀνώλεθρον, ὥσπερ φησὶν Ἀναξίμανδρος καὶ οἱ πλεῖστοι τῶν φυσιολόγων. Nor is it without reason that they all make it a principle or source. We cannot say that the infinite has no effect, and the only effectiveness which we can ascribe to it is that of a principle. Everything is either a source or derived from a source. But there cannot be a source of the infinite or limitless, for that would be a limit of it. Further, as it is a beginning, it is both uncreatable and indestructible. For there must be a point at which what has come to be reaches completion, and also a termination of all passing away. That is why, as we say, there is no principle of this, but it is this which is held to be the principle of other things, and to encompass all and to steer all, as those assert who do not recognize, alongside the infinite, other causes, such as Mind or Friendship. Further they identify it with the Divine, for it is 'deathless and imperishable' as Anaximander says, with the majority of the physicists.
Postquam philosophus determinavit de motu, hic incipit determinare de infinito. 326. After settling motion, the Philosopher now begins to settle the infinite. Et primo ostendit quod ad scientiam naturalem pertinet determinare de infinito; secundo incipit determinare, ibi: esse autem infinitum et cetera. First he shows that natural science should settle the infinite; Secondly, he begins to determine the infinite, at 336. Circa primum duo facit: primo ostendit quod ad scientiam naturalem pertinet determinare de infinito; secundo ponit opiniones antiquorum philosophorum de infinito, ibi: et omnes tanquam principium et cetera. As to the first he does two things: First he shows that it pertains to natural science to settle the infinite. Secondly, he gives the opinions of the earlier philosophers concerning the infinite, at 329. Primum ostendit et ratione et signo. 327. He proves the first point with an argument and a sign. Ratio talis est. Scientia naturalis consistit circa magnitudines et tempus et motum; sed necesse est finitum aut infinitum in his inveniri: omnis enim magnitudo vel motus vel tempus sub altero horum continetur, id est sub finito vel infinito; ergo ad naturalem philosophum pertinet considerare de infinito, an sit et quid sit. The argument is as follows: Natural science studies magnitudes and time and motion. But in such things the finite and infinite are necessarily found for every magnitude or motion or time is contained under one or the other, i.e., either the finite or the infinite. Therefore, it pertains to natural science to consider the infinite, namely, as to whether it exists and as to what it is. Sed quia posset aliquis dicere quod consideratio de infinito pertinet ad philosophum primum ratione suae communitatis, ad hoc excludendum interponit quod non omne ens oportet esse finitum vel infinitum: nam punctus et passio, idest passibilis qualitas, sub nullo horum continetur: ea autem quae pertinent ad considerationem philosophi primi, consequuntur ens inquantum ens est, et non aliquod determinatum genus entis. But because it could be objected that consideration of the infinite pertains to first philosophy, on account of its general character, he counters this by saying that not every being has to be either finite or infinite; for a point and a passion, i.e., passible [sensible] quality, are not contained under either, whereas the objects of consideration in first philosophy are things that follow upon being inasmuch as it is being and not upon some definite genus of being. Deinde cum dicit: signum enim quod huius scientiae etc., ostendit idem per signum acceptum a consideratione philosophorum naturalium. Omnes enim qui rationabiliter tractaverunt huiusmodi philosophiam, scilicet naturalem, fecerunt mentionem de infinito. Ex quo colligitur probabile argumentum ab auctoritate sapientum, quod ad philosophiam naturalem pertineat determinare de infinito. 328. Then  he establishes the same point through a sign taken from the practice of the natural philosophers. For all who have treated this, namely, natural philosophy, according to reason, have mentioned the infinite. This fact is a probable argument, based on the authority of wise men, that it belongs to natural philosophy to settle the infinite. Deinde cum dicit: et omnes tanquam principium etc., ponit opiniones antiquorum de infinito. 329. Then  he gives the opinion of the earlier philosophers about the infinite. Et primo ostendit in quo diversificabantur; secundo ostendit in quo omnes conveniebant, ibi: rationabiliter autem et cetera. First he shows in what they differ; Secondly, he shows in what they all agreed, at 335. Circa primum duo facit: primo ponit opiniones philosophorum non naturalium de infinito, scilicet Pythagoricorum et Platonicorum; secundo opiniones naturalium, ibi: qui autem de natura omnes et cetera. As to the first he does two things: First he gives the opinions on the infinite of those philosophers who were non-natural [i.e., disregarded sense], i.e., the Pythagoreans, and the Platonists; Secondly, he gives the opinions of the natural philosophers, at 333. Circa primum duo facit: primo ostendit in quo conveniebant Pythagorici et Platonici; secundo in quo differebant, ibi: praeter hoc quod Pythagorici et cetera. As to the first he does two things: First he dhows the points of agreement between the Pythagoreans and the Platonists, secondly, their points of disagreement, at 331. Dicit ergo primo quod omnes philosophi posuerunt infinitum esse sicut quoddam principium entium; sed hoc fuit proprium Pythagoricis et Platonicis, quod ponerent infinitum non esse accidens alicui alteri naturae, sed esse quoddam per se existens. Et hoc competebat eorum opinioni, quia ponebant numeros et quantitates esse substantias rerum; infinitum autem in quantitate est; unde et infinitum per se existens ponebant. 330. He says therefore that while all the philosophers posited the infinite as a certain principle of things, only the Pythagoreans and Platonists asserted that the infinite is not something accidental to some nature but something existing of itself. This is not surprising, because it is in keeping with their claim that numbers and quantities are the substances of things. Now the infinite is found in quantity; hence they posited that the infinite exists of itself. Deinde cum dicit: praeter hoc quod Pythagorici etc., ostendit differentiam inter Platonem et Pythagoricos: et primo quantum ad positionem infiniti; secundo quantum ad radicem ipsius, ibi: et hi quidem infinitum esse et cetera. 331. Then  he shows the difference between Plato and the Pythagorean, first, as to the laying down of the infinite; secondly, as to the basis thereof (no. 332). Quantum autem ad positionem infiniti, in duobus differebat Plato a Pythagoricis. Pythagorici enim non ponebant infinitum nisi in sensibilibus: cum enim infinitum competat quantitati, prima autem quantitas est numerus, Pythagorici non ponebant numerum separatum a sensibilibus, sed dicebant numerum esse substantiam rerum sensibilium; et per consequens neque infinitum erat nisi in sensibilibus. Regarding the laying down of the infinite, Plato differed in two respects from the Pythagoreans. For the Pythagoreans did not lay down an infinite except in sensible things. Since the infinite belongs to quantity, and the first quantity is number, the Pythagoreans, not laying down number to be separated from sensible things, but stating number to be, rather, the substance of sensible things, consequently did not lay down any infinite except in sensible things. Item Pythagoras considerabat quod sensibilia quae sunt infra caelum, sunt circumclausa caelo, unde in eis non potest esse infinitum: et propter hoc ponebat quod infinitum esset in sensibilibus extra caelum. Likewise Pythagoras considered that the sensible beings which are within the confines of the heavens are circumscribed by the heavens—whence the infinite cannot be in them—hence he laid down that the infinite was in the sensible things outside the heavens, Sed Plato e contrario ponebat quod nihil est extra caelum: neque enim dicebat esse extra caelum aliquod corpus sensibile, quia caelum dicebat esse continens omnia sensibilia; neque etiam ideas et species rerum, quas ponebat esse separatas, dicebat esse extra caelum, quia intus et extra significant locum; ideae vero secundum ipsum non sunt in aliquo loco, quia locus corporalium est. But Plato by contrast laid down that nothing is outside the heavens. For neither did he say that there was outside the heavens any sensible body, since he maintained that the heavens contained all sensible things; nor did he say that the ideas and species of things, which he laid down as being separated, were outside the heavens, since “inside of” and “outside of” signify place, while the ideas, according to him, are not in any place, place being of corporeal things. Item dicebat Plato quod infinitum non solum est in rebus sensibilibus, sed etiam in illis, idest in ideis separatis; quia etiam in ipsis numeris separatis est aliquid formale, ut unum, et aliquid materiale, ut duo, ex quibus omnes numeri componuntur. Plato likewise said that the infinite is not only in sensible things, but also in “them”, i.e., the separated ideas, there being, even in the separated numbers something formal, such as unity, and something material, such as duality, out of which all numbers are composed. Deinde cum dicit: et hi quidem infinitum esse parem etc., ostendit differentiam eorum quantum ad radicem infiniti. 332. Then  he shows the difference between them as to the basis of the infinite. Et dicit quod Pythagorici attribuebant infinitum uni radici, scilicet numero pari. Et hoc manifestabant dupliciter. Primo per rationem: quia id quod concluditur ab alio et per aliud terminatur, quantum est de se, habet rationem infiniti; quod autem concludit et terminat, habet rationem termini. Par autem numerus comprehenditur et concluditur sub impari. Si enim proponitur aliquis numerus par, undique divisibilis apparet; cum vero addita unitate ad imparem numerum reducitur, iam quandam indivisionem consequitur, ac si par sub impari constringatur: unde videtur quod par sit per se infinitum, et causet in aliis infinitatem. And he says that the Pythagoreans attributed the infinite to a basis which was “even number.” And they demonstrated this in two ways. The first was an argument. That which is enclosed by another, and is terminated by another has the nature of the infinite; whereas that which encloses and terminated has the nature of a term. Now even number is comprehended and included under odd number. For if some even number is proposed, it is seen as in every way divisible. But when by the addition of unity it is reduced to an odd number, it now takes on a certain indivisibility, as though even was compressed under odd. Hence it seems as though “even” is infinite in itself, and causes infinity in others. Ostendit etiam idem per signum. Ad cuius evidentiam sciendum est quod in geometricis, gnomon dicitur quadratum super diametrum consistens cum duobus supplementis: huiusmodi igitur gnomon circumpositus quadrato, constituit quadratum. Ex huius ergo similitudine in numeris gnomones dici possunt numeri qui aliquibus numeris adduntur. Secondly, the same is shown by an example. To follow it one must know that in geometry a “gnomon” is the name for a square on the diameter with two supplements [i.e., three squares put together to form the shape of an “L”]. If a square is added to this gnomon, a square is constituted. From this likeness those numbers may be called “gnomons” which are added to certain numbers. Est autem hic observandum, quod si aliquis accipiat numeros impares secundum ordinem progressionis naturalis, et unitati, quae est quadratum virtute (inquantum semel unum est unum), addat primum numerum imparem, scilicet ternarium, constituetur quaternarius, qui est numerus quadratus; nam bis duo sunt quatuor. Si vero huic secundo quadrato addatur secundus impar scilicet quinarius, consurgit novenarius, qui est quadratum ternarii; nam ter tria sunt novem. Si autem huic tertio quadrato addatur tertius impar, scilicet septenarius, consurgit sedecim, qui est quadratum quaternarii: et sic semper per ordinatam additionem numerorum imparium resultat eadem forma in numeris, scilicet quadratum. Here one should notice that if one takes the odd numbers according to the order of natural progression, and to unity, which is a square as to power (since one times one is one), one adds the first odd number, namely, three, there will be constituted four, which is a squared number since twice two is four. If now to this second square there is added the second odd number, namely, five, one obtains nine, which is the square of three, since three times three is nine. Then if to this third square there is added the third odd number, namely, seven, one obtains sixteen, which is the square of four. And thus, following the ordered addition of odd numbers, there always arises the same form in those numbers, namely, a square. Per additionem autem parium, semper resultat diversa figura. Nam si primus par, scilicet duo, addantur unitati, consurgit ternarius, qui est figurae trilaterae; si autem huic addatur secundus par, scilicet quaternarius, consurgit septenarius, qui est figurae heptagonae: et sic semper variatur figura numerorum ex additione parium. By the addition of even numbers, however, there is always produced a different shape. For if the first even number, namely, two, be added to unity, there arises three, which has a triangular figure; if then to this there be added the second even number, namely, four, one has seven, which is in the shape of a heptagon. And thus, in this wise the figure of the resulting numbers constantly varies with the addition of even numbers. Et hoc videtur esse signum quod uniformitas pertinet ad numerum imparem, difformitas autem et varietas et infinitum pertinent ad numerum parem. And this appears to be a sign that uniformity belongs to odd number, while difformity and variation and the infinite belong to even number. Et hoc est quod dicit: signum huius, scilicet quod infinitum sequatur numerum parem, est hoc quod contingit in numeris: circumpositis enim gnomonibus, idest numeris additis, circa unum, idest circa unitatem, et extra, idest circa alios numeros, aliquando quidem fit alia species, idest alia forma numeralis, scilicet per additionem numeri paris; aliquando autem fit una species, scilicet per additionem numeri imparis. Et sic patet quare Pythagoras numero pari attribuerit infinitatem. Hence he says, namely, that a sign of this, i.e., that infinity follows even number, is what occurs in numbers. For by the addition of gnomons, i.e., numbers, to one, i.e., to unity, and outside, i.e., to other numbers, sometimes there occurs another species, i.e., another natural form, namely, when one adds an even number; sometimes there occurs a single species, namely, when one adds an odd number. From this it is evident why Pythagoras attributed infinity to even number. Plato autem attribuebat duabus radicibus, scilicet magno et parvo: haec enim duo secundum ipsum sunt ex parte materiae, cui competit infinitum. But Plato attributed it to two roots, namely, to the “large” and the “small.” Tor these two, according to him, belong to matter, to which in turn the infinite belongs. Deinde cum dicit: qui autem de natura etc., ponit opiniones naturalium philosophorum de infinito. Sciendum est ergo quod omnes naturales philosophi, qui scilicet naturaliter principia rerum tradiderunt, dixerunt quod infinitum non est per se subsistens, sicut supra dictum est; sed ponunt infinitum esse accidens alicuius naturae ei suppositae. Qui ergo posuerunt unum principium tantum materiale, quodcumque sit, de numero eorum quae dicuntur elementa, sive aer sive aqua sive aliquid medium, dixerunt illud esse infinitum. Qui vero fecerunt plura elementa sed finita secundum numerum, nullus eorum posuit quod elementa essent infinita secundum quantitatem: ipsa enim distinctio elementorum contrariari videbatur infinitati utriusque eorum. Sed illi qui fecerunt infinita secundum numerum, dicunt ex omnibus illis infinitis fieri quoddam unum infinitum per contactum. 333. Then  he gives the opinions of the natural philosophers about the infinite. He says that all the natural philosophers, those, namely, who gave natural [i.e., sensible principles for things, taught that the infinite does not subsist by itself, as said above (no. 330), but is an accident of some nature. Hence those who posited just one material principle (some member of the list of things called elements, i.e., air or water or something intermediate) said it was infinite. But of those who posited a finite number of principles, none supposed them to be infinite in quantity: for the very distinction of the elements seemed to conflict with the notion that they could be infinite. But those who posited. an infinitude of principles said that from all those infinites was formed one infinite through contact. Et hi fuerunt Anaxagoras et Democritus: qui in duobus differebant. 334. Those who taught this were Anaxagoras and Democritus, who differed in two respects. Primo quidem in quidditate principiorum infinitorum: nam Anaxagoras posuit illa infinita principia esse infinitas similes partes, ut carnis et ossis et huiusmodi; Democritus autem posuit huiusmodi infinita principia esse indivisibilia corpora, differentia secundum figuras; quae quidem corpora dicebat esse semina totius naturae. They differed first as to the nature of the infinite principles: for Anaxagoras taught that the infinite principles were infinite similar parts of flesh and of bone and so on; but Democritus taught that-they were indivisible bodies differing in figure. He said these bodies were the seeds of all of nature. Alia differentia est quantum ad habitudinem horum principiorum ad invicem. Anaxagoras enim dixit quod quaelibet harum partium infinitarum esset commixta cuilibet, sicut quod in qualibet parte carnis esset os et e converso, et similiter de aliis. Et hoc ideo, quia vidit quod quodlibet fit ex quolibet; et cum crederet quod omne quod fit ex aliquo, est in eo, syllogizavit quod quodlibet sit in quolibet. Et ex hoc videtur ipse affirmare quod aliquando omnes res erant simul confusae ad invicem, et nihil erat distinctum ab alio. Sicut enim haec caro et hoc os commiscentur ad invicem, quod demonstratur per generationem eorum ad invicem, sic etiam est de quolibet alio. Omnia igitur aliquando fuerunt simul. Est enim accipere principium disgregationis non solum in aliquo uno, sed in omnibus simul: quod sic probabat. Quod enim fit ex alio, erat prius ei commixtum, et per hoc fit, quod segregatur ab eo; sed omnia fiunt, licet non simul; oportet igitur ponere unum principium generationis omnium, non solum uniuscuiusque. Et hoc unum principium vocavit intellectum, cui soli competit distinguere et congregare, propter hoc quod est immixtus. Another difference was as in the relation of these principles one to the other. For Anaxagoras said that each of these parts was a mixture of all the others, so that in each part of flesh there was bone and vice versa and the same for the other parts. He came to this opinion because he saw that anything came from anything; and, hence, since he believed that whatever comes to be from something is in it, he concluded that everything is in everything. And from this he seems to assert that at some time all things were commingled and nothing was distinct from anything else. Just as this flesh and this bone are commingled (which is proved by the generation of one from the other) so is everything else commingled. Therefore at one time all things were together. For it is necessary to posit a principle of separation not only in one single thing but in all things simultaneously. He proved this thus: Whatever comes to be from something other was previously commingled with it and is produced by being separated from it; but all things are produced, though not all at the same time; therefore, there must be some one principle generating not only each thing but all things. This one principle he called “intellect,” which alone has the capacity to separate and bring together because it is itself uncommingled. Quod autem fit per intellectum, videtur habere quoddam principium; quia intellectus a determinato principio incipiens operatur. Si ergo segregatio fit ab intellectu, oportet dicere quod segregatio habeat quoddam principium; unde concludebat quod aliquando omnia fuerint simul, et quod motus quo segregantur res ab invicem, aliquando incoeperit, cum prius non fuerit. Sic igitur Anaxagoras posuit unum principium fieri ex altero. Now whatever comes to be through intellect seems to have a principle; because intellect acts by starting from a definite principle. Therefore, if separation is brought about by intellect, separation must have a principle; hence, he concluded, at some time all things were together and the motion by which things were separated one from the other began in time, and did not previously exist. Thus Anaxagoras laid down one principle as producing another. Sed Democritus dicit quod unum principium non fit ex altero: sed tamen natura corporis, quae est communis omnibus indivisibilibus corporibus, differens secundum partes et figuras, est principium omnium secundum magnitudinem, inquantum ex indivisibilibus ponebat componi omnes magnitudines divisibiles. But Democritus said that one principle is not derived from another, but that the nature of body which is common to all indivisible bodies, though different in parts and figure, is the principle of all things according to magnitude, for he posited that all divisible magnitudes are composed of indivisibles. Et sic concludit quod ad philosophum naturalem pertinet considerare de infinito. And thus does Aristotle conclude that to consider the infinite pertains to the natural philosopher. Deinde cum dicit: rationabiliter autem et principium etc., ponit quatuor, in quibus antiqui philosophi concordabant circa infinitum. Quorum primum est, quod omnes posuerunt infinitum esse principium; et hoc rationabiliter, idest per probabilem rationem. Non enim possibile est, si infinitum est, quod sit frustra, idest quod non habeat aliquem determinatum gradum in entibus. Nec potest habere aliam virtutem nisi principii: quia omnia quae sunt in mundo, vel sunt principia vel ex principiis; infinito autem non competit habere principium, quia quod habet principium, habet finem. Unde relinquitur quod infinitum sit principium. Sed attendendum est quod in hac ratione utuntur aequivoce principio et fine: nam quod est ex principio, habet principium originis; infinito autem repugnat principium et finis quantitatis vel magnitudinis. 335. Then  he outlines four points of agreement among the early philosophers in regard to the infinite. The first of which is that all posited the infinite as a principle, and this “reasonably” i.e., for the following reason: If the infinite exists, it is impossible for It to be in vain, i.e., that it lack some definite standing among the beings of reality. But it can have no power other than that of a principle. For all things in the world are either principles, or derived from principles. But it is not fitting for the infinite to have a principle, because what has a principle has an end. Hence it follows that the infinite is a principle. Note, however, that in this reasoning, “principle” and “end” are both used equivocally; for that which is derived from a principle has a principle of origin whereas it is to have a principle and end of its quantity or size which is incompatible with infinite. Secundum autem quod attribuebant infinito est, quod sit ingenitum et incorruptibile. Et hoc sequitur ex eo quod est principium. Omne enim quod fit, necesse est quod accipiat finem, sicut et habet principium; et etiam cuiuslibet corruptionis est aliquis finis: finis autem repugnat infinito; unde esse generabile et corruptibile repugnat infinito. Et sic patet quod non est aliquod principium infiniti, sed magis infinitum est principium aliorum. The second point of agreement is that they denied coming into existence and ceasing to exist to the infinite. This follows from the fact that it is a principle. For whatever is produced must have an ending just as it has a principle; and likewise any process of corruption has an end. But “end” and “infinite” are incompatible; hence the infinite can neither be generated nor corrupted. Hence it is clear that the infinite has no principle, but that the infinite is the principle of everything else. Et in hoc etiam aequivoce sumebant principium et finem, sicut et supra. This argument, too, uses “principle” and “end” equivocally, as above. Tertium autem quod attribuebant infinito erat, quod contineret et gubernaret omnia: hoc enim videtur esse primi principii. Et hoc dixerunt quicumque non posuerunt praeter materiam, quam dicebant infinitam, alias causas, scilicet agentes, ut intellectum posuit Anaxagoras et concordiam Empedocles. Continere enim et gubernare magis pertinet ad principium agens, quam ad materiam. The third point of agreement is that they attributed to the infinite the prerogative of containing and governing all things, for this seems to belong to a first principle. And this was the opinion of those who did not grant in addition to matter, which they said was infinite, other causes, namely, agent causes, as Anaxagoras posited an intellect and Empedocles concord. For to contain and to govern pertain more to an active principle than to matter. Quartum autem quod infinito attribuebant est, quod esset quoddam divinum: omne enim quod est immortale aut incorruptibile, divinum appellabant: et hoc posuit Anaximander et plures antiquorum philosophorum naturalium. The fourth point of agreement was to attribute divinity to the infinite; for whatever is immortal or incorruptible they called divine. This was the doctrine of Anaximander and a number of the ancient natural philosophers.
Arguments for and against the infinite
Chapter 4 cont. τοῦ δ' εἶναί τι ἄπειρον ἡ πίστις ἐκ πέντε μάλιστ' ἂν συμβαίνοι σκοποῦσιν, Belief in the existence of the infinite comes mainly from five considerations:
- ἔκ τε τοῦ χρόνου (οὗτος γὰρ ἄπειρος)
- καὶ ἐκ τῆς ἐν τοῖς μεγέθεσι διαιρέσεως (χρῶνται γὰρ καὶ οἱ μαθηματικοὶ τῷ ἀπείρῳ)·
- ἔτι τῷ οὕτως ἂν μόνως μὴ ὑπολείπειν γένεσιν καὶ φθοράν, εἰ ἄπειρον εἴη ὅθεν ἀφαιρεῖται τὸ γιγνόμενον·
- ἔτι τῷ τὸ πεπερασμένον ἀεὶ πρός τι περαίνειν, ὥστε ἀνάγκη μηδὲν εἶναι πέρας, εἰ ἀεὶ περαίνειν ἀνάγκη ἕτερον πρὸς ἕτερον.
- μάλιστα δὲ καὶ κυριώτατον, ὃ τὴν κοινὴν ποιεῖ ἀπορίαν πᾶσι· διὰ γὰρ τὸ ἐν τῇ νοήσει μὴ ὑπολείπειν καὶ ὁ ἀριθμὸς δοκεῖ ἄπειρος εἶναι καὶ τὰ μαθηματικὰ μεγέθη καὶ τὸ ἔξω τοῦ οὐρανοῦ.
- From the nature of time—for it is infinite.
- From the division of magnitudes—for the mathematicians also use the notion of the infinite.
- If coming to be and passing away do not give out, it is only because that from which things come to be is infinite.
- Because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself.
- Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody—not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought.
ἀπείρου δ' ὄντος τοῦ ἔξω, καὶ σῶμα ἄπειρον εἶναι δοκεῖ καὶ κόσμοι· τί γὰρ μᾶλλον τοῦ κενοῦ ἐνταῦθα ἢ ἐνταῦθα; ὥστ' εἴπερ μοναχοῦ, καὶ πανταχοῦ εἶναι τὸν ὄγκον. ἅμα δ' εἰ καὶ ἔστι κενὸν καὶ τόπος ἄπειρος, καὶ σῶμα εἶναι ἀναγκαῖον· ἐνδέχεσθαι γὰρ ἢ εἶναι οὐδὲν διαφέρει ἐν τοῖς ἀϊδίοις. The last fact (that what is outside is infinite) leads people to suppose that body also is infinite, and that there is an infinite number of worlds. Why should there be body in one part of the void rather than in another? Grant only that mass is anywhere and it follows that it must be everywhere. Also, if void and place are infinite, there must be infinite body too, for in the case of eternal things what may be must be. ἔχει δ' ἀπορίαν ἡ περὶ τοῦ ἀπείρου θεωρία· καὶ γὰρ μὴ εἶναι τιθεμένοις πόλλ' ἀδύνατα συμβαίνει καὶ εἶναι. ἔτι δὲ ποτέρως ἔστιν, πότερον ὡς οὐσία ἢ ὡς συμβεβηκὸς καθ' αὑτὸ φύσει τινί; ἢ οὐδετέρως, ἀλλ' οὐδὲν ἧττον ἔστιν ἄπειρον ἢ ἄπειρα (204a.) τῷ πλήθει; μάλιστα δὲ φυσικοῦ ἐστιν σκέψασθαι εἰ ἔστι μέγεθος αἰσθητὸν ἄπειρον. But the problem of the infinite is difficult: many contradictions result whether we suppose it to exist or not to exist. If it exists, we have still to ask how it exists; as a substance or as the essential attribute of some entity? Or in neither way, yet none the less is there something which is infinite or some things which are infinitely many? The problem, however, which specially belongs to the physicist is to investigate whether there is a sensible magnitude which is infinite. πρῶτον οὖν διοριστέον ποσαχῶς λέγεται τὸ ἄπειρον.
- ἕνα μὲν δὴ τρόπον τὸ ἀδύνατον διελθεῖν τῷ μὴ πεφυκέναι διιέναι, ὥσπερ ἡ φωνὴ ἀόρατος·
- ἄλλως δὲ τὸ διέξοδον ἔχον ἀτελεύτητον, ἢ ὃ μόγις,
- ἢ ὃ πεφυκὸς ἔχειν μὴ ἔχει διέξοδον ἢ πέρας. ἔτι ἄπειρον ἅπαν ἢ κατὰ πρόσθεσιν ἢ κατὰ διαίρεσιν ἢ ἀμφοτέρως.
We must begin by distinguishing the various senses in which the term 'infinite' is used.
- What is incapable of being gone through, because it is not in its nature to be gone through (the sense in which the voice is 'invisible').
- What admits of being gone through, the process however having no termination, or what scarcely admits of being gone through.
- What naturally admits of being gone through, but is not actually gone through or does not actually reach an end. Further, everything that is infinite may be so in respect of addition or division or both.
Chapter 5 Χωριστὸν μὲν οὖν εἶναι τὸ ἄπειρον τῶν αἰσθητῶν, αὐτό τι ὂν ἄπειρον, οὐχ οἷόν τε. εἰ γὰρ μήτε μέγεθός ἐστιν μήτε πλῆθος, ἀλλ' οὐσία αὐτό ἐστι τὸ ἄπειρον καὶ μὴ συμβεβη κός, ἀδιαίρετον ἔσται (τὸ γὰρ διαιρετὸν ἢ μέγεθος ἔσται ἢ πλῆθος)· εἰ δὲ τοιοῦτον, οὐκ ἄπειρον, εἰ μὴ ὡς ἡ φωνὴ ἀόρατος. ἀλλ' οὐχ οὕτως οὔτε φασὶν εἶναι οἱ φάσκοντες εἶναι τὸ ἄπειρον οὔτε ἡμεῖς ζητοῦμεν, ἀλλ' ὡς ἀδιεξίτητον. εἰ δὲ κατὰ συμβεβηκὸς ἔστιν τὸ ἄπειρον, οὐκ ἂν εἴη στοιχεῖον τῶν ὄντων, ᾗ ἄπειρον, ὥσπερ οὐδὲ τὸ ἀόρατον τῆς διαλέκτου, καίτοι ἡ φωνή ἐστιν ἀόρατος. Now it is impossible that the infinite should be a thing which is itself infinite, separable from sensible objects. If the infinite is neither a magnitude nor an aggregate, but is itself a substance and not an attribute, it will be indivisible; for the divisible must be either a magnitude or an aggregate. But if indivisible, then not infinite, except in the sense (1) in which the voice is 'invisible'. But this is not the sense in which it is used by those who say that the infinite exists, nor that in which we are investigating it, namely as (2) 'that which cannot be gone through'. But if the infinite exists as an attribute, it would not be, qua infinite an element in substances, any more than the invisible would be an element of speech, though the voice is invisible. ἔτι πῶς ἐνδέχεται εἶναί τι αὐτὸ ἄπειρον, εἴπερ μὴ καὶ ἀριθμὸν καὶ μέγεθος, ὧν ἐστι καθ' αὑτὸ πάθος τι τὸ ἄπειρον; ἔτι γὰρ ἧττον ἀνάγκη ἢ τὸν ἀριθμὸν ἢ τὸ μέγεθος. Further, how can the infinite be itself any thing, unless both number and magnitude, of which it is an essential attribute, exist in that way? If they are not substances, a fortiori the infinite is not. φανερὸν δὲ καὶ ὅτι οὐκ ἐνδέχεται εἶναι τὸ ἄπειρον ὡς ἐνεργείᾳ ὂν καὶ ὡς οὐσίαν καὶ ἀρχήν· ἔσται γὰρ ὁτιοῦν αὐτοῦ ἄπειρον τὸ λαμβανόμενον, εἰ μεριστόν (τὸ γὰρ ἀπείρῳ εἶναι καὶ ἄπειρον τὸ αὐτό, εἴπερ οὐσία τὸ ἄπειρον καὶ μὴ καθ' ὑποκειμένου), ὥστ' ἢ ἀδιαίρετον ἢ εἰς ἄπειρα διαιρετόν· πολλὰ δ' ἄπειρα εἶναι τὸ αὐτὸ ἀδύνατον (ἀλλὰ μὴν ὥσπερ ἀέρος ἀὴρ μέρος, οὕτω καὶ ἄπειρον ἀπείρου, εἴ γε οὐσία ἐστὶ καὶ ἀρχή)· ἀμέριστον ἄρα καὶ ἀδιαίρετον. ἀλλ' ἀδύνατον τὸ ἐντελεχείᾳ ὂν ἄπειρον· ποσὸν γάρ τι εἶναι ἀναγκαῖον. κατὰ συμβεβηκὸς ἄρα ὑπάρχει τὸ ἄπειρον. ἀλλ' εἰ οὕτως, εἴρηται ὅτι οὐκ ἐνδέχεται αὐτὸ λέγειν ἀρχήν, ἀλλ' ᾧ συμβέβηκε, τὸν ἀέρα ἢ τὸ ἄρτιον. ὥστε ἀτόπως ἂν ἀποφαίνοιντο οἱ λέγοντες οὕτως ὥσπερ οἱ Πυθαγόρειοί φασιν· ἅμα γὰρ οὐσίαν ποιοῦσι τὸ ἄπειρον καὶ μερίζουσιν. ἀλλ' ἴσως αὕτη μὲν [ἐστι] καθόλου ἡ ζήτησις, εἰ ἐνδέχεται ἄπειρον καὶ ἐν τοῖς μαθηματικοῖς (204b.) εἶναι καὶ ἐν τοῖς νοητοῖς καὶ μηδὲν ἔχουσι μέγεθος· ἡμεῖς δ' ἐπισκοποῦμεν περὶ τῶν αἰσθητῶν καὶ περὶ ὧν ποιούμεθα τὴν μέθοδον, ἆρ' ἔστιν ἐν αὐτοῖς ἢ οὐκ ἔστι σῶμα ἄπειρον ἐπὶ τὴν αὔξησιν. It is plain, too, that the infinite cannot be an actual thing and a substance and principle. For any part of it that is taken will be infinite, if it has parts: for 'to be infinite' and 'the infinite' are the same, if it is a substance and not predicated of a subject. Hence it will be either indivisible or divisible into infinites. But the same thing cannot be many infinites. (Yet just as part of air is air, so a part of the infinite would be infinite, if it is supposed to be a substance and principle.) Therefore the infinite must be without parts and indivisible. But this cannot be true of what is infinite in full completion: for it must be a definite quantity. Suppose then that infinity belongs to substance as an attribute. But, if so, it cannot, as we have said, be described as a principle, but rather that of which it is an attribute—the air or the even number. Thus the view of those who speak after the manner of the Pythagoreans is absurd. With the same breath they treat the infinite as substance, and divide it into parts. This discussion, however, involves the more general question whether the infinite can be present in mathematical objects and things which are intelligible and do not have extension, as well as among sensible objects. Our inquiry (as physicists) is limited to its special subject-matter, the objects of sense, and we have to ask whether there is or is not among them a body which is infinite in the direction of increase.
Positis opinionibus antiquorum de infinito, hic incipit inquirere veritatem. 336. Having listed the opinions of the earlier philosophers on the infinite, Aristotle now begins to inquire into the truth of the matter. Et primo obiicit ad utramque partem; secundo solvit, ibi: quod quidem igitur actu corpus et cetera. First he objects to both sides of the question; Secondly, he solves the objections, in Lecture 10. Circa primum duo facit: primo ponit rationes ad ostendendum quod infinitum sit; secundo ad ostendendum quod non sit, ibi: habet autem dubitationem et cetera. About the first he does two things: First he gives reasons to show that the infinite exists; Secondly, to show that it does not exist, at 342. Circa primum ponit quinque rationes. 337. In regard to the first he gives five reasons to show that the infinite exists. Quarum prima sumitur ex tempore, quod secundum communem opinionem antiquorum infinitum erat: solus enim Plato generavit tempus, ut in octavo huius dicetur. The first of these is taken from time, which, according to the common opinion of the ancients, was infinite. Indeed, Plato alone generated time, as will be said in Book VIII (l.2). Dicit ergo primo quod ad ostendendum infinitum esse, ex quinque rationibus accipi potest: et primo quidem ex tempore, quod est infinitum secundum illos qui dicebant tempus semper fuisse et semper futurum esse. He says therefore first that the infinite is shown to exist by five arguments. Abe first of these is taken from time, which is infinite according to those who held that time always was and always will be. Secunda ratio sumitur ex divisione magnitudinum in infinitum. Infinito enim in magnitudinibus utuntur etiam mathematici in suis demonstrationibus: quod non esset si infinitum totaliter tolleretur a rebus: oportet igitur ponere infinitum. 338. The second reason is taken from the infinite divisibility of magnitude. For even mathematicians use the infinite in their demonstrations. This, however, would not happen, if there were no infinite at all; hence the infinite exists. Tertia ratio sumitur ex perpetuitate generationis et corruptionis, secundum plurium opinionem. Si enim totaliter tolleretur infinitum, non posset dici quod generatio et corruptio in infinitum durarent; unde oporteret dicere quod quandoque totaliter generatio cessaret, quod est contra multorum opinionem. Oportet igitur ponere infinitum. 339. The third reason is based on the perpetual processes of generation and. corruption according to the opinion of many; for if the infinite were denied, generation and corruption could not endure indefinitely; hence, it would have to be admitted that generation would sometime cease, which is against the opinion of many. Therefore, it is necessary to posit the infinite. Quarta ratio sumitur ex apparenti ratione finiti. Videtur enim pluribus quod de ratione finiti sit, quod semper includatur ab aliquo alio: quia videmus apud nos omne finitum extendi usque ad aliquid. Demonstrato igitur aliquo corpore, si illud sit infinitum, habetur propositum; si autem sit finitum, oportebit quod terminetur ad aliquid aliud, et iterum illud, si sit finitum, ad aliquid aliud. Aut ergo erit procedere in infinitum, aut devenietur ad aliquod corpus infinitum; et utroque modo ponitur infinitum. Unde necesse est quod nullus sit terminus corporum, si semper oportet quod omne finitum includatur ab aliquo altero. 340. The fourth reason is based on the apparent nature of the infinite, to many seems to consist in this that it is something always included by something else, because we observe that every finite reaches into something else. Let a body be pointed out; if it be infinite, then the infinite exists; if it be finite, it must be terminated at something else, and this latter, if it in turn be finite, at something else. We must either proceed thus to infinity or come to a body that is infinite. In either case, the infinite exists. Hence there can be no end to bodies, if every finite body is always included by some other. Quinta ratio sumitur ab apprehensione intellectus vel imaginationis. Unde dicit quod illud quod maxime facit communem dubitationem inducentem homines ad ponendum infinitum, est ex hoc, quod intellectus nunquam deficit, quin super quodlibet finitum datum possit aliquid addere. Existimabant autem antiqui philosophi quod res responderent apprehensioni intellectus et sensus: unde dicebant quod omne quod videtur, est verum, ut dicitur in IV Metaphys.: et propter hoc credebant quod etiam in rebus esset infinitum. Inde est enim quod videtur numerus esse infinitus: quia intellectus cuilibet numero dato unitatem addendo, facit aliam speciem. Et eadem ratione videntur magnitudines mathematicae, quae in imaginatione consistunt, esse infinitae: quia qualibet magnitudine data, possumus imaginari maiorem. Et eadem ratione videtur esse extra caelum quoddam spatium infinitum: quia possumus imaginari extra caelum in infinitum quasdam dimensiones. 341. The fifth reason is taken from the apprehension of the intellect or of the imagination. Hence, he says that that which chiefly constitutes the common difficulty which induces men to posit the infinite is that the intellect never is exhausted but can always add something to any given finite amount. Now the earlier philosophers supposed that things corresponded to the intellect’s or senses’ apprehension of them; hence because they said that whatever appeared to be is true, as stated in Metaphysics IV (l.11), they believed that even in reality there exists an infinite. Hence number seems to be infinite, because the intellect can always create a new number, simply by adding unity to a given number. For the same reason mathematical magnitudes, which exist in the imagination, seem to be infinite, because, given any definite magnitude, we can imagine a greater. And for the same reason there seems to be an infinite space beyond the heavens, because we can imagine certain dimensions existing beyond the heavens to infinity. Si autem est infinitum spatium extra caelum, necesse videtur quod sit corpus infinitum, et quod sint mundi infiniti. Et hoc duplici ratione. Prima ratio est, quia si consideretur totum spatium infinitum, totum secundum se consideratum est uniforme: non est ergo assignare rationem quare magis in una parte illud spatium sit vacuum a corpore quam in alia. Si ergo in aliqua parte illius spatii invenitur magnitudo corporalis huius mundi, oportet quod in qualibet parte illius spatii inveniatur aliqua magnitudo corporalis sicut quae est huius mundi: et sic oportet corpus esse infinitum sicut et spatium: vel etiam oportet mundos esse infinitos, ut Democritus posuit. Alia ratio est ad idem ostendendum; quia si est infinitum spatium, aut est vacuum aut est plenum. Si est plenum, habetur propositum, quod sit corpus infinitum: si autem est vacuum, cum vacuum nihil aliud sit quam locus non repletus corpore, possibilis tamen repleri, necesse est quod si est spatium infinitum, sit etiam locus infinitus, qui possit repleri corpore. Et ita oportebit esse corpus infinitum, quia in perpetuis non differt contingere et esse. Unde si contingit locum infinitum repleri corpore, oportet dicere quod sit repletus corpore infinito. Necesse ergo videtur dicere quod sit corpus infinitum. Now if there be infinite space beyond the heavens, it seems that there is an infinite body and even infinite worlds. This for two reasons. The first is that if the totality of space be considered infinite, that totality will be uniform; hence, there is no reason why that space should be devoid of body in one part rather than in another, Therefore, if there is found in one part of that space the bodily magnitude of this world, then there should be found in each part of that space some bodily magnitude comparable to that of this world. Thus body must be infinite in the same way as space or there must even exist infinite worlds, as Democritus supposed. Another reason proving the same point is that if there be infinite space, it is either empty or full. If it is full, we have our point that there is infinite body; but if it is empty, then since the empty is a place not filled with a body but capable of being so filled, it follows that if space is infinite, there is infinite place capable of being filled with body. Thus there must be infinite body, because in perpetual matters, there is no difference between what can be and what is. Hence, if infinite place can be filled with body, it must be admitted that it is filled with infinite body. Therefore, it seems necessary to say that there is infinite body. Deinde cum dicit: habet autem dubitationem etc., obiicit in contrarium. Et circa hoc tria facit. 342. Then  he takes the opposite position. And in regard to this he does three things: Primo ostendit quaestionem esse dubitabilem, ne rationes praemissae omnino verum concludere videantur; secundo ostendit quot modis dicitur infinitum, ibi: primum ergo determinandum etc.; tertio ponit rationes ad ostendendum infinitum non esse, ibi: separabile quidem igitur esse et cetera. First, he shows that the matter is debatable, lest anyone suppose that the afore -mentioned reasons are unassailable; Secondly, he gives the various meanings of the word “infinite,” at 344; Thirdly, he gives reasons showing that the infinite does not exist, at 345. Dicit ergo primo quod dubitatio est circa infinitum, utrum sit vel non sit: multa enim impossibilia consequuntur iis qui ponunt infinitum omnino non esse, sicut ex praemissis patet; et etiam iis qui ponunt infinitum esse, multa accidunt impossibilia, ut ex consequentibus rationibus patebit. Est etiam dubitatio qualiter infinitum sit, utrum scilicet sit aliquid per se existens, sicut quaedam substantia; vel sicut aliquod accidens per se conveniens alicui naturae; aut neutro modo sit (scilicet neque per se existens, sicut substantia, neque sicut accidens per se), sed nihilominus, si est accidens, est aliquod infinitum continuum, et aliqua infinita secundum multitudinem. Sed maxime pertinet ad considerationem philosophi naturalis, si est aliqua magnitudo sensibilis infinita: nam magnitudo sensibilis est magnitudo naturalis. 343. He says therefore  that there is a question about whether the infinite exists or not. For, on the one hand, many impossibilities follow upon holding that it does not; those, for example, listed in 337 ff. On the other hand, there are also difficulties attendant upon holding that the infinite does exist, as will be clear subsequently (no. 345 ff). There is doubt also as to its manner of existence. Does it exist as a substance does, or as an accident belonging essentially to some nature? Of if neither as a substance nor as an essential accident, but as an accident nevertheless, is there some infinite continuum and are there things infinite in number? Now it very much pertains to the philosopher of nature to discuss whether there exists such a thing as an infinite sensible magnitude, for a sensible magnitude is a natural magnitude. Deinde cum dicit: primum ergo determinandum est etc., ostendit quot modis dicitur infinitum: et ponit duas divisiones infiniti. Quarum prima est communis infinito et omnibus privative dictis. 344. Then  he shows in how many ways “infinite” is said, and lists two divisions of the infinite. The first division is con on to the infinite and to all things said privatively. Nam invisibile dicitur tripliciter, vel quod non est aptum natum videri, ut vox, quae non est de genere visibilium; vel quod male videtur, sicut quod videtur in obscuro aut a remotis; vel quod natum est videri et non videtur, sicut quod est omnino in tenebris. Sic igitur et uno modo dicitur infinitum, quod non est natum transiri (nam infinitum idem est quod intransibile): et hoc est quia est de genere intransibilium, sicut indivisibilia ut punctus et forma; per quem etiam modum dicitur vox invisibilis. Alio modo dicitur infinitum, quod quantum est de se, transiri potest, sed eius transitus non potest perfici a nobis, sicut si dicatur profunditas maris esse infinita: vel si potest perfici, tamen vix et cum difficultate, sicut si dicamus quod iter usque in Indiam est infinitum. Et utrumque istorum pertinet ad hoc quod est esse male transibile. Tertio modo dicitur infinitum, quod est natum transiri quasi de genere transibilium existens, quod tamen non habet transitum ad finem; ut si esset aliqua linea non habens terminum, vel quaecumque alia quantitas: et sic proprie dicitur infinitum. For “invisible” is said in three ways: either as denoting 1) what of its very nature 13 not apt to be seen, for example, a sound which is not in the genus of visible things; or 2) what is difficult to see, as what is seen in the dark or from a distance; 3) what is apt to be seen but is not, as something in total darkness. Correspondingly, what of its very nature is not apt to be passed over is called “infinite” (for the infinite is the same as that which cannot be passed over)—and this is because it belongs to the genus of intraversable things, as are indivisibles, such as a point and a form; this is the way that a sound was called invisible. In a second way, infinity is ascribed to what could be passed over but its passage is impossible for us; thus, we say that the depth of the sea is infinite; or if it could be passed through, it would be with difficulty, as if we should say that a trip to India is infinite. Both of these belong to that which is “difficultly traversable.” In a third way, infinity is ascribed to what is passable but there is no passage to its terminus; for example, a line without an end or any other such quantity without limits; this is the proper cense of the word “infinite.” Aliam divisionem propriam infiniti ponit ibi: amplius infinitum etc., dicens quod infinitum dicitur vel per appositionem, sicut in numeris; aut secundum divisionem, sicut in magnitudinibus; aut utroque modo, sicut in tempore. He then gives the other division of infinite, [233 bis], saying that infinity is spoken either by addition, as in numbers, or according to division, as in magnitudes, or both ways, as in time. Deinde cum dicit: separabile quidem igitur etc., ponit rationes ad excludendum infinitum: et primo ad excludendum infinitum separatum, quod Platonici posuerunt; secundo ad excludendum infinitum a rebus sensibilibus, ibi: rationabiliter quidem igitur et cetera. 345. Then  he lays down the arguments leading to an exclusion of the infinite: First those excluding a separated infinite, such as laid down by the Platonists; Secondly, those excluding the infinite from sensible things, at no. 349. Circa primum ponit tres rationes. Circa quarum primam dicit quod impossibile est infinitum esse separatum a sensibilibus, ita quod ipsum infinitum sit aliquid per se existens, sicut Platonici posuerunt. Quia si ponitur infinitum esse aliquid separatum, aut habet aliquam quantitatem (scilicet continuam quae est magnitudo, aut discretam quae est multitudo), aut non. Si est substantia sine accidente quod est magnitudo vel multitudo, oportet quod infinitum sit indivisibile: quia omne divisibile vel est numerus vel magnitudo. Si autem aliquid est indivisibile, non erit infinitum nisi primo modo, scilicet prout dicitur aliquid infinitum quod non est aptum natum transiri, sicut dicitur vox invisibilis: sed hoc est praeter intentionem praesentis quaestionis, qua quaerimus de infinito, et praeter intentionem eorum qui posuerunt infinitum; non enim intenderunt ponere infinitum sicut indivisibile, sed sicut intransibile, idest quod natum est transiri et non habet transitum. With respect to the first he lays down three reasons. As to the first of these he says that it is impossible for the infinite to be separated from sensible things, in such a way that the infinite should be something existing of itself, as the Platonists laid down. For if the infinite is laid down as something separated, either it has a certain quantity (namely, continuous, which is size, or discrete, which is number), or not. If it is a substance without either the accident of size or that of number, then the infinite must be indivisible—since whatever is divisible is either number or size. But if something is indivisible, it will not be infinite except in the first way, namely, as something is called “infinite” which is not by nature susceptible to being passed through, in the same way that a sound is said to be “invisible” [as not being by nature susceptible to being seen], but this is not what is intended in the present inquiry concerning the infinite, nor by those who laid down the infinite. For they did not intend to lay down the infinite as something indivisible, but as something that could not be passed through, i.e., as being susceptible to such, but with the passage having no completion. Si vero infinitum non sit solum substantia, sed etiam habeat accidens quod est magnitudo et multitudo cui competit infinitum, et sic infinitum insit substantiae secundum illud accidens; non erit infinitum inquantum huiusmodi principium eorum quae sunt, sicut antiqui posuerunt; sicut etiam non dicimus invisibile esse principium locutionis, quamvis accidat voci, quae est principium locutionis. If, however, the infinite should not only be a substance, but also should have an accident which is size or number to which the infinite belongs, in such a way that the infinite would be inherent in the substance in the manner of that accident, then the principle of existing things will not be infinite as such the ancients laid down, just as we do not say that the principle of speech is invisible, although such a thing is an accident of sound, which is the principle of speech. Secundam rationem ponit ibi: amplius quomodo contingit etc.: et est talis. Minus est separabile et per se existens passio quam subiectum; sed infinitum est passio magnitudinis et numeri; sed magnitudo et numerus non possunt separari et per se existere, ut in metaphysica probatum est; ergo neque infinitum. 346. The second reason  is as follows. A passion is less separable and able to exist of itself than a subject. But the infinite is a passion of size and number—which cannot be separated and exist of themselves, as is proved in the Metaphysics [XI, l.10]. Therefore neither can this be so of the infinite. Tertiam rationem ponit ibi: manifestum autem est et cetera. Et dicit manifestum esse quod non potest poni, quod infinitum sit in actu, et quod sit sicut substantia quaedam, et sicut principium rerum. Aut enim infinitum erit partibile, aut impartibile. 347. The third argument  is as follows. He [Aristotle] states that it is clear that the infinite cannot be laid down as being in act, and as being a certain substance, and as being the principle of things. For the infinite is either divisible, or indivisible. Si quidem erit partibile, necesse est quod quaelibet pars eius sit infinitum, si infinitum est substantia: quia si infinitum est substantia, et non dicitur de aliquo subiecto ut accidens, oportebit quod idem sit infinitum et infinito esse, idest essentia et ratio infiniti. Non enim idem est id quod est album et natura albi: sed id quod est homo, est hoc quod est natura hominis. Unde oportebit quod si infinitum sit substantia, aut sit indivisibile, aut dividatur in partes infinitas, quod est impossibile; quia ex multis infinitis componi aliquid idem est impossibile, quia oporteret infinitum terminari ad aliud infinitum. If indeed it is divisible, every one of its parts will have to be infinite, on the supposition that the infinite is a substance. For if it is a substance, and is not predicated of any subject as an accident, then that which is infinite and the nature of the infinite, i.e., the essence and notion of the infinite, will have to be the same. For that which is white and the nature of white are not the same, but that which is man, and the nature of man, are. Whence it will be necessary that the infinite, if it be a substance, be either indivisible, or divided into parts which are infinite—which is impossible, since it is impossible to compose some same thing out of many infinities, as this would involve one infinite’s being terminated by another infinite. Apparet etiam non solum ex ratione sed etiam ex similitudine, quod si infinitum sit substantia et dividatur, oportet quod quaelibet pars eius sit infinita. It likewise appears not only from argument but also from an analogy that if the infinite is a substance and is divided, it is necessary that each and every part of it be infinite. Sicut enim quaelibet pars aeris est aer, ita et quaelibet pars infiniti erit infinita, si infinitum sit substantia et principium. Quia si sit principium, oportet infinitum esse substantiam simplicem, non compositam ex partibus difformibus, sicut homo, cuius non quaelibet pars est homo. Cum ergo impossibile sit alicuius infiniti quamlibet partem esse infinitam, oportet quod infinitum sit impartibile et indivisibile. Sed illud quod est indivisibile, non potest esse infinitum in actu: quia quod est infinitum in actu est quantum, et omne quantum est divisibile. Sequitur ergo quod si est infinitum in actu, non sit sicut substantia, sed sub ratione accidentis quod est quantitas. Et si hoc sit infinitum, non erit principium, sed illud cui accidit infinitum; sive illud sit aliqua substantia sensibilis, ut aer, sicut posuerunt philosophi naturales; sive sit aliqua substantia intelligibilis, ut par, sicut posuerunt Pythagorici. For just as every part of air is air, so too every part of the infinite will be infinite, if the infinite is a substance and a principle. For if it is a principle, the infinite has to be a simple substance, not composed out of differing parts, as in the case of man whose every part is not man. Since, therefore, it is impossible for every part of some infinite to be infinite, the infinite must then be unable to be reduced to parts, and indivisible. But what is indivisible cannot be infinite in act—since whatever is infinite in act is quantified, and everything quantified is divisible. It follows, therefore, that if there be any infinite in act, it is not after the manner of substance, but has the reason of the accident which is quantity. And if this be infinite, it will not be a principle, but that to which the infinite occurs, whether it be some sensible substance, such as air; or some intelligible substance, such as “even,” as the Pythagoreans laid down. Unde manifestum est quod inconvenienter dixerunt Pythagorici, ponentes infinitum esse substantiam, et simul cum hoc ponentes ipsum esse divisibile: quia sequitur quod quaelibet pars eius sit infinita; quod est impossibile, ut supra dictum est. Whence it is plain that the Pythagoreans did not speak sensibly, positing the infinite as a substance, and the same time holding it as divisible—since it follows that every part of it would be infinite, which is impossible, as said above. Ultimo autem dicit quod ista quaestio, quae est: an infinitum sit in mathematicis quantitatibus et in rebus intelligibilibus non habentibus magnitudinem, est magis universalis quam sit praesens consideratio. Nos enim intendimus ad praesens de rebus sensibilibus, de quibus tradimus scientiam naturalem: utrum in ipsis sit corpus infinitum in augmentum, ut antiqui naturales posuerunt. 348. Finally, he says that this question “whether there be an infinite in mathematical quantities and in intelligible things not having magnitude” is a more general one than the present question. For our question concerns sensible things about which natural science treats: “Whether among natural things there be a body infinite in size, such as the early philosophers posited.
No Sensible Infinite
Chapter 5 cont. λογικῶς μὲν οὖν σκοπουμένοις ἐκ τῶν τοιῶνδε δόξειεν ἂν οὐκ εἶναι· εἰ γάρ ἐστι σώματος λόγος τὸ ἐπιπέδῳ ὡρισμένον, οὐκ ἂν εἴη σῶμα ἄπειρον, οὔτε νοητὸν οὔτε αἰσθητόν (ἀλλὰ μὴν οὐδ' ἀριθμὸς οὕτως ὡς κεχωρισμένος καὶ ἄπειρος· ἀριθμητὸν γὰρ ἀριθμὸς ἢ τὸ ἔχον ἀριθμόν· εἰ οὖν τὸ ἀριθμητὸν ἐνδέχεται ἀριθμῆσαι, καὶ διεξελθεῖν ἂν εἴη δυνατὸν τὸ ἄπειρον)· We may begin with a dialectical argument and show as follows that there is no such thing. If 'bounded by a surface' is the definition of body there cannot be an infinite body either intelligible or sensible. Nor can number taken in abstraction be infinite, for number or that which has number is numerable. If then the numerable can be numbered, it would also be possible to go through the infinite. φυσικῶς δὲ μᾶλλον θεωροῦσιν ἐκ τῶνδε. οὔτε γὰρ σύνθετον οἷόν τε εἶναι οὔτε ἁπλοῦν. σύνθετον μὲν οὖν οὐκ ἔσται τὸ ἄπειρον σῶμα, εἰ πεπερασμένα τῷ πλήθει τὰ στοιχεῖα. ἀνάγκη γὰρ πλείω εἶναι, καὶ ἰσάζειν ἀεὶ τἀναντία, καὶ μὴ εἶναι ἓν αὐτῶν ἄπειρον (εἰ γὰρ ὁποσῳοῦν λείπεται ἡ ἐν ἑνὶ σώματι δύναμις θατέρου, οἷον εἰ τὸ πῦρ πεπέρανται, ὁ δ' ἀὴρ ἄπειρος, ἔστιν δὲ τὸ ἴσον πῦρ τοῦ ἴσου ἀέρος τῇ δυνάμει ὁποσαπλασιονοῦν, μόνον δὲ ἀριθμόν τινα ἔχον, ὅμως φανερὸν ὅτι τὸ ἄπειρον ὑπερβαλεῖ καὶ φθερεῖ τὸ πεπερασμένον)· ἕκαστον δ' ἄπειρον εἶναι ἀδύνατον· σῶμα μὲν γάρ ἐστιν τὸ πάντῃ ἔχον διάστασιν, ἄπειρον δὲ τὸ ἀπεράντως διεστηκός, ὥστε τὸ ἄπειρον σῶμα πανταχῇ ἔσται διεστηκὸς εἰς ἄπειρον. ἀλλὰ μὴν οὐδὲ ἓν καὶ ἁπλοῦν εἶναι σῶμα ἄπειρον ἐνδέχεται, οὔτε ὡς λέγουσί τινες τὸ παρὰ τὰ στοιχεῖα, ἐξ οὗ ταῦτα γεννῶσιν, οὔθ' ἁπλῶς. εἰσὶν γάρ τινες οἳ τοῦτο ποιοῦσι τὸ ἄπειρον, ἀλλ' οὐκ ἀέρα ἢ ὕδωρ, ὅπως μὴ τἆλλα φθείρηται ὑπὸ τοῦ ἀπείρου αὐτῶν· ἔχουσι γὰρ πρὸς ἄλληλα ἐναντίωσιν, οἷον ὁ μὲν ἀὴρ ψυχρός, τὸ δ' ὕδωρ ὑγρόν, τὸ δὲ πῦρ θερμόν· ὧν εἰ ἦν ἓν ἄπειρον, ἔφθαρτο ἂν ἤδη τἆλλα· νῦν δ' ἕτερον εἶναί φασιν ἐξ οὗ ταῦτα. ἀδύνατον δ' εἶναι τοιοῦτον, οὐχ ὅτι ἄπειρον (περὶ τούτου μὲν γὰρ κοινόν τι λεκτέον ἐπὶ παντὸς ὁμοίως, καὶ ἀέρος καὶ ὕδατος καὶ ὁτουοῦν), ἀλλ' ὅτι οὐκ ἔστιν τοιοῦτον σῶμα αἰσθητὸν παρὰ τὰ καλούμενα στοιχεῖα· ἅπαντα γὰρ ἐξ οὗ ἐστι, καὶ διαλύεται εἰς τοῦτο, ὥστε ἦν ἂν ἐνταῦθα παρὰ ἀέρα καὶ πῦρ καὶ γῆν καὶ ὕδωρ· φαίνεται δ' οὐδέν. οὐδὲ δὴ πῦρ οὐδ' ἄλλο τι (205a.) τῶν στοιχείων οὐδὲν ἄπειρον ἐνδέχεται εἶναι. ὅλως γὰρ καὶ χωρὶς τοῦ ἄπειρον εἶναί τι αὐτῶν, ἀδύνατον τὸ πᾶν, κἂν ᾖ πεπερασμένον, ἢ εἶναι ἢ γίγνεσθαι ἕν τι αὐτῶν, ὥσπερ Ἡράκλειτός φησιν ἅπαντα γίγνεσθαί ποτε πῦρ (ὁ δ' αὐτὸς λόγος καὶ ἐπὶ τοῦ ἑνός, οἷον ποιοῦσι παρὰ τὰ στοιχεῖα οἱ φυσικοί)· πάντα γὰρ μεταβάλλει ἐξ ἐναντίου εἰς ἐναντίον, οἷον ἐκ θερμοῦ εἰς ψυχρόν. If, on the other hand, we investigate the question more in accordance with principles appropriate to physics, we are led as follows to the same result. The infinite body must be either (1) compound, or (2) simple; yet neither alternative is possible. (1) Compound the infinite body will not be, if the elements are finite in number. For they must be more than one, and the contraries must always balance, and no one of them can be infinite. If one of the bodies falls in any degree short of the other in potency—suppose fire is finite in amount while air is infinite and a given quantity of fire exceeds in power the same amount of air in any ratio provided it is numerically definite—the infinite body will obviously prevail over and annihilate the finite body. On the other hand, it is impossible that each should be infinite. 'Body' is what has extension in all directions and the infinite is what is boundlessly extended, so that the infinite body would be extended in all directions ad infinitum. Nor (2) can the infinite body be one and simple, whether it is, as some hold, a thing over and above the elements (from which they generate the elements) or is not thus qualified. (a) We must consider the former alternative; for there are some people who make this the infinite, and not air or water, in order that the other elements may not be annihilated by the element which is infinite. They have contrariety with each other—air is cold, water moist, fire hot; if one were infinite, the others by now would have ceased to be. As it is, they say, the infinite is different from them and is their source. It is impossible, however, that there should be such a body; not because it is infinite on that point a general proof can be given which applies equally to all, air, water, or anything else—but simply because there is, as a matter of fact, no such sensible body, alongside the so-called elements. Everything can be resolved into the elements of which it is composed. Hence the body in question would have been present in our world here, alongside air and fire and earth and water: but nothing of the kind is observed. (b) Nor can fire or any other of the elements be infinite. For generally, and apart from the question of how any of them could be infinite, the All, even if it were limited, cannot either be or become one of them, as Heraclitus says that at some time all things become fire. (The same argument applies also to the one which the physicists suppose to exist alongside the elements: for everything changes from contrary to contrary, e.g. from hot to cold).
Postquam philosophus removit opinionem antiquorum qui de infinito non naturaliter loquebantur, illud a sensibilibus separantes, hic ostendit non esse infinitum, sicut philosophi naturales ponebant. 349. After rejecting the opinion of the earlier philosophers who spoke non-naturally of the infinite, separating it from sensible things, the Philosopher now shows there is no infinite even in the sense in which the natural philosophers laid it down. Et primo ostendit hoc per rationes logicas; secundo per rationes naturales, ibi: physice autem magis et cetera. First he shows this by logical reasons; Secondly by natural reasons in 353. Dicuntur autem primae rationes logicae, non quia ex terminis logicis logice procedant, sed quia modo logico procedunt, scilicet ex communibus et probabilibus, quod est proprium syllogismi dialectici. The first set of reasons are called “logical,” not because they proceed logically from logical terms, but because they proceed in a logical manner, i.e., from common and probable propositions, which is the characteristic of the dialectical syllogism. Ponit ergo duas logicas rationes. In quarum prima ostenditur quod non sit aliquod corpus infinitum. Definitio enim corporis est, quod sit determinatum planitie, idest superficie, sicut definitio lineae est quod eius termini sint puncta. Nullum autem corpus determinatum superficie, est infinitum: ergo nullum corpus est infinitum; neque sensibile, quod est corpus naturale, neque intelligibile, quod est corpus mathematicum. Quod ergo dicit rationabiliter, exponendum est logice: nam logica dicitur rationalis philosophia. 350. He gives therefore  two logical reasons. In the first of these it is shown that there is no infinite body. For the definition of body is that it is determined by a surface, just as the definition of a line is that its terms are points. But no body determined by a surface is infinite. Therefore, no body is infinite, whether it be sensible, i.e., a natural body, or intelligible, i.e., a mathematical body. (The word “rational” [or dialectical] should be here expounded as “logical” indeed, logic is called “rational philosophy.”) Secunda ratio ostendit quod non sit infinitum multitudine. Omne enim numerabile contingit numerari, et per consequens numerando transiri; omnis autem numerus, et omne quod habet numerum, est numerabile; ergo omne huiusmodi contingit transiri. Si igitur aliquis numerus, sive separatus, sive in sensibilibus existens, sit infinitus, sequetur quod possibile sit transire infinitum; quod est impossibile. 351. The second reason shows that there is no infinite multitude. For everything countable can be numbered and consequently passed through by counting. But every number and whatever has a number is countable. Therefore, every such thing can be passed over. If, therefore any number, whether separated or existing in sensible things, be infinite, it follows that the infinite can be passed through, which is impossible. Attendendum est autem quod istae rationes sunt probabiles, et procedentes ex iis quae communiter dicuntur. Non enim ex necessitate concludunt: quia qui poneret aliquod corpus esse infinitum, non concederet quod de ratione corporis esset terminari superficie, nisi forte secundum potentiam; quamvis hoc sit probabile et famosum. Similiter qui diceret aliquam multitudinem esse infinitam, non diceret eam esse numerum, vel numerum habere. Addit enim numerus super multitudinem rationem mensurationis: est enim numerus multitudo mensurata per unum, ut dicitur in X Metaphys. Et propter hoc numerus ponitur species quantitatis discretae, non autem multitudo; sed est de transcendentibus. 352. Notice that these reasons are probable and proceed from common premises. For they do not conclude of necessity: in effect, whoever posits an infinite body would not concede that it would of its very nature be terminated by a surface, except perhaps potentially; although this is probable and well-known. Similarly, whoever would posit an infinite multitude would not admit it to be a number or that it has a number. For number adds to multitude the notion of measure, because a number is “multitude measured by unity,” as is said in Metaphysics X. For this reason number is considered to be a species of discrete quantity, but multitude is not; it is, rather, a transcendental. Deinde cum dicit: physice autem magis etc., inducit rationes naturales ad ostendendum quod non sit corpus infinitum in actu. 353. Then  he produces natural reasons to show that there is no infinite body in act. Circa quas considerandum est quod, quia Aristoteles nondum probaverat corpus caeleste esse alterius essentiae a quatuor elementis, opinio autem communis suo tempore fuerat quod esset de natura quatuor elementorum, procedit in his rationibus ac si non esset aliud corpus sensibile extra quatuor elementa, secundum suam consuetudinem: quia semper antequam probet id quod est suae opinionis, procedit ex suppositione opinionis aliorum communis. Unde postquam probavit in primo libro de caelo et mundo, caelum esse alterius naturae ab elementis, ad veritatis certitudinem iterat considerationem de infinito, ostendens universaliter quod nullum corpus sensibile est infinitum. In connection with these reasons one must consider that since Aristotle had not yet proved that the heavenly body was of another essence from that of the four elements, and the common opinion of his time was that it was of the same nature as the four elements, he therefore proceeds in these reasonings as though there were no other sensible body outside of the four elements. This is in keeping with his custom, since he always, before proving that which is his own belief, proceeds from what is supposed by the common opinion of others. Hence, after he proved in De Caelo I (l.4) that the heavens are of another nature from the elements, he repeats, for the sake of the certitude of the truth, the consideration of the infinite, showing unqualifiedly that no sensible body is infinite. Hic autem primo ostendit quod non sit corpus sensibile infinitum, supposito quod sint elementa finita multitudine; secundo ostendit idem universaliter, ibi: oportet autem de omni et cetera. Here, however, he first shows that there is no sensible infinite body on the supposition that the elements are finite in number; secondly he shows the same thing in a universal way, at no. 358. Dicit ergo primo quod procedendo naturaliter, idest ex principiis scientiae naturalis, magis et certius considerari poterit quod non sit corpus sensibile infinitum, ex iis quae dicentur. Omne enim corpus sensibile aut est simplex aut compositum. He says therefore first that when one proceeds “naturally,” i.e., according to the principles of natural science, one is better able, and with more certitude, to consider that there is no sensible infinite body from what will be said. For every sensible body is either simple or composite. Primo ergo ostendit quod non sit corpus sensibile compositum infinitum, supposito quod sint elementa finita secundum multitudinem. Non enim potest esse quod unum ipsorum sit infinitum et alia finita: quia ad compositionem alicuius corporis mixti requiritur quod sint plura elementa, et quod contraria aliquo modo adaequentur; alias compositio permanere non posset; quia illud quod esset omnino potentius, destrueret alia, cum elementa sint contraria. Si autem unum elementorum esset infinitum, nulla aequalitas esset, aliis finitis existentibus; quia infinitum improportionaliter excedit finitum. Non ergo hoc potest esse, quod unum tantum eorum quae veniunt in mixtionem, sit infinitum. 354. First therefore he shows that there is no composite sensible body that is infinite, supposing that the elements are finite according to multitude. For it cannot be that one of the elements is infinite and the others finite—because the composition of any compound body requires that there be a number of elements and that the contraries therein be somehow in equilibrium. If this were not so, the composition could not endure—for the strongest would destroy all the others, since the elements are contrary. But if one of the elements were infinite, no equilibrium would ensue as long as the other elements were finite, because there is no proportion between infinite and finite. Therefore it cannot be that only one of the elements in the composite be infinite. Posset autem aliquis dicere quod illud infinitum esset debilis virtutis in agendo, et ideo non potest vincere alia, scilicet finita, quae sunt fortioris virtutis, utpote si infinitus sit aer et finitus ignis. Et ideo ad hoc removendum dicit, quod quantumcumque potentia unius corporis quod ponitur infinitum, deficiat a potentia alterius corporis quod ponitur finitum, utpote si ignis sit finitus et aer infinitus; necesse est tamen dicere quod aer quantumcumque duplicatus, idest secundum aliquem numerum multiplicatus, sit aequalis igni in potentia. Si enim potentia ignis est centuplo maior quam potentia aeris eiusdem quantitatis, si aer centuplicetur secundum quantitatem, erit aequalis ei in potentia: et tamen aer centuplicatus est multiplicatus secundum aliquem numerum determinatum, et vincitur a potentia totius aeris infiniti. Unde manifestum est quod etiam potentia ignis vincetur a potentia aeris infiniti; et sic infinitum excellit et corrumpit finitum, quantumcumque potentioris naturae videatur. But someone could claim that the infinite element might have such weak energy in acting, that it would not destroy the finite elements which are stronger; for example, if the infinite one were air and the finite one fire. And therefore, to remove this objection he says that no matter how much less the energy of that one infinite body is than that of the finite body (for example, if fire be infinite and air finite) nevertheless an infinite accumulation of air would be equal in energy to the fire. For if the energy of the fire is one hundred times greater than an equal quantity of air, then if the air be multiplied a hundredfold, it will equal the fire in energy; and yet air multiplied a hundred times is multiplied according to a finite number and is exceeded by the power of the whole infinite amount of air. Hence, it is clear that even the energy of the fire will be overcome by the energy of infinite air: thus the infinite will excel and corrupt the finite, no matter how powerful its nature. Similiter etiam non potest esse quod quodlibet elementorum ex quibus componitur corpus mixtum, sit infinitum: quia de ratione corporis est quod habeat dimensiones in omnem partem, non in longitudinem tantum ut linea, neque in longitudinem et latitudinem solum ut superficies: de ratione autem infiniti est, quod habeat distantias seu dimensiones infinitas; ergo de ratione corporis infiniti est quod habeat dimensiones infinitas in omnem partem. Et sic non potest esse quod ex pluribus corporibus infinitis aliquod unum componatur, quia quodlibet occupat totum mundum; nisi ponantur duo corpora esse simul, quod est impossibile. 355. Similarly, it cannot be that any of the elements out of which a compound body is composed be infinite; because it is a property of a body that it have dimensions in every direction, and not in length only, as in a line, or in length and width only, as on a surface. But the nature of the infinite is to have infinite “distances” or dimensions. Therefore, the infinite body should have infinite dimensions in every direction. Thus, it cannot be that one body result from a number of infinite bodies, because each occupies the whole world, unless you posit that two bodies interpenetrate, which is impossible. Sic igitur ostenso quod corpus compositum non potest esse infinitum, ostendit ulterius quod nec etiam corpus simplex, neque unum elementorum, neque aliquod medium inter ea, ut vapor est medium inter aerem et aquam. Quidam enim posuerunt hoc esse principium, ex eo alia generari dicentes. Et hoc dicebant esse infinitum: non autem aerem, aut aquam, aut aliquod aliorum elementorum; quia contingeret alia elementa corrumpi a quocumque ipsorum, quod infinitum poneretur, quia elementa habent contrarietatem ad invicem, cum aer sit humidus, aqua frigida, ignis calidus, terra sicca: unde si unum horum esset infinitum, corrumperet alia, cum contrarium natum sit corrumpi a contrario. Et ideo dicunt aliquid aliud ab elementis esse infinitum, ex quo sicut ex principio elementa causantur. 356. Therefore, having shown that a composite body cannot be infinite, he now proves that neither can a simple body, nor one of the elements, nor any medium among the elements(taking vapor as a medium between air and water) be infinite. For some posited this last as a principle stating other things to be generated from it. And they said that this was something infinite, but not air or water or any of the other elements; because the other elements would be corrupted by whichever one was supposed as infinite. For the elements have contrariety one to the other since air is humid, water cold, fire hot and earth dry. Hence if one of them were infinite, it would destroy the others, since one contrary is disposed to be corrupted by another. And that is why they said that something other than the elements was infinite, from which, as from a principle, the elements arose. Hanc autem positionem dicit esse impossibilem, non solum quantum ad hoc, quod dicit tale corpus medium esse infinitum, quia de hoc dicetur communis quaedam ratio tam de igne et aere, et aqua, quam etiam de corpore medio; sed ex hoc ipso etiam est impossibilis praedicta positio, quia ponit aliquod principium elementare praeter quatuor elementa. Now he states this position to be impossible not only as to its maintaining such a mediate body to be infinite, since there will be applied a same common argument [in no. 35] to fire and air and water and likewise to the mediate body, but also as to its laying down some elemental principle in addition to the elements. Non enim invenitur aliquod corpus sensibile praeter ea quae dicuntur elementa, scilicet aerem, aquam et huiusmodi: sed hoc oporteret si aliquid aliud praeter elementa veniret in compositionem istorum corporum. Unumquodque enim compositum resolvitur in ea ex quibus componitur. Si igitur aliquid aliud veniret in compositionem istorum corporum quam haec quatuor elementa, sequeretur quod hic apud nos inveniretur aliquod corpus simplex praeter ista elementa, per resolutionem istorum in elementa. Sic igitur patet quod positio praemissa falsa est quantum ad hoc, quod posuit aliquod corpus simplex praeter haec elementa nota. For there is found no sensible body outside of those things called the “elements,” namely, air, water, and the like. But this would have to be the case if anything besides the elements should enter into the composition of such bodies. If, therefore, anything else should enter into the composition of those bodies in addition to the four elements, it would follow that we should find here some simple body besides the elements, by the resolution of the above bodies into their elements. It follows therefore that the aforesaid position is false as to its positing of some simple body besides the known elements. Ulterius autem ostendit communi ratione, quod nullum elementorum possit esse infinitum: quia si aliquod elementorum esset infinitum, impossibile esset totum universum esse aliud nisi illud elementum; et oporteret quod omnia alia elementa converterentur in ipsum, vel iam essent conversa in ipsum, propter excellentiam virtutis infiniti super alia: sicut Heraclitus dicit quod quandoque futurum est quod omnia convertantur in ignem, propter excellentem ignis virtutem. Et eadem ratio est de uno elementorum et de alio corpore quod faciunt quidam naturales extra elementa. Oportet enim illud aliud habere contrarietatem ad elementa, cum ex eo ponantur alia generari: mutatio autem non fit nisi ex contrario in contrarium, ut ex calido in frigidum, sicut supra ostensum est. Sic igitur et istud corpus medium ratione contrarietatis destruet alia elementa. 357. He further shows by a general argument that none of the elements can be infinite. For if any of the elements were infinite, it would be impossible for the whole universe to be anything but that element. It would likewise be necessary that all the other elements be changed into it, or to have already been changed into it, due to the excess of power of the infinite over other things, as Heraclitus says that at some future time all things will be converted into fire because of the excelling power of fire. And the same reason holds good for one of the elements and for come other body that some natural philosophers create besides the elements. For it is necessary that this other body have contrariety toward the elements, since other things are laid down as being generated from it, and change does not take place except from;)ne contrary to another, as in the case of going from hot to cold, as shown above (I, l.10). This middle body would therefore in this way destroy, by reason of its contrariety, the other elements.
No infinite body shown absolutely
Chapter 5 cont. δεῖ δὲ κατὰ παντὸς ἐκ τῶνδε σκοπεῖν, εἰ ἐνδέχεται ἢ οὐκ ἐνδέχεται εἶναι [σῶμα ἄπειρον αἰσθητόν]. ὅτι δὲ ὅλως ἀδύνατον εἶναι σῶμα ἄπειρον αἰσθητόν, ἐκ τῶνδε δῆλον. The preceding consideration of the various cases serves to show us whether it is or is not possible that there should be an infinite sensible body. The following arguments give a general demonstration that it is not possible. πέφυκε γὰρ πᾶν τὸ αἰσθητόν που εἶναι, καὶ ἔστιν τόπος τις ἑκάστου, καὶ ὁ αὐτὸς τοῦ μορίου καὶ παντός, οἷον ὅλης τε τῆς γῆς καὶ βώλου μιᾶς, καὶ πυρὸς καὶ σπινθῆρος. It is the nature of every kind of sensible body to be somewhere, and there is a place appropriate to each, the same for the part and for the whole, e.g. for the whole earth and for a single clod, and for fire and for a spark. ὥστε εἰ μὲν ὁμοειδές, ἀκίνητον ἔσται ἢ ἀεὶ οἰσθήσεται· καίτοι ἀδύνατον (τί γὰρ μᾶλλον κάτω ἢ ἄνω ἢ ὁπουοῦν; λέγω δὲ οἷον, εἰ βῶλος εἴη, ποῦ αὕτη κινηθήσεται ἢ ποῦ μενεῖ; ὁ γὰρ τόπος ἄπειρος τοῦ συγγενοῦς αὐτῇ σώματος. πότερον οὖν καθέξει τὸν ὅλον τόπον; καὶ πῶς; τίς οὖν ἢ ποῦ ἡ μονὴ καὶ ἡ κίνησις αὐτῆς; ἢ πανταχοῦ μενεῖ; οὐ κινηθήσεται ἄρα. ἢ πανταχοῦ κινηθήσεται; οὐκ ἄρα στήσεται)· εἰ δ' ἀνόμοιον τὸ πᾶν, ἀνόμοιοι καὶ οἱ τόποι· καὶ πρῶτον μὲν οὐχ ἓν τὸ σῶμα τοῦ παντὸς ἀλλ' ἢ τῷ ἅπτεσθαι· ἔπειτα ἤτοι πεπερασμένα ταῦτ' ἔσται ἢ ἄπειρα τῷ εἴδει. πεπερασμένα μὲν οὖν οὐχ οἷόν τε (ἔσται γὰρ τὰ μὲν ἄπειρα τὰ δ' οὔ, εἰ τὸ πᾶν ἄπειρον, οἷον τὸ πῦρ ἢ τὸ ὕδωρ· φθορὰ δὲ τὸ τοιοῦτον τοῖς ἐναντίοις [καθάπερ εἴρηται πρότερον · (καὶ διὰ τοῦτ' οὐθεὶς τὸ ἓν καὶ ἄπειρον πῦρ ἐποίησεν οὐδὲ γῆν τῶν φυσιολόγων, ἀλλ' ἢ ὕδωρ ἢ ἀέρα ἢ τὸ μέσον αὐτῶν, ὅτι τόπος ἑκατέρου δῆλος ἦν διωρισμένος, ταῦτα δ' ἐπαμφοτερίζει τῷ ἄνω καὶ κάτω.) εἰ δ' ἄπειρα καὶ ἁπλᾶ, καὶ οἱ τόποι ἄπειροι, καὶ ἔσται ἄπειρα τὰ στοιχεῖα· εἰ δὲ τοῦτ' ἀδύνατον καὶ πεπερασμένοι οἱ τόποι, καὶ τὸ ὅλον [πεπεράνθαι ἀναγκαῖον]· ἀδύνατον γὰρ μὴ ἀπαρτίζειν τὸν τόπον καὶ τὸ σῶμα· οὔτε γὰρ ὁ τόπος ὁ πᾶς μείζων ἢ ὅσον ἐνδέχεται τὸ σῶμα εἶναι (ἅμα δ' οὐδ' ἄπειρον ἔσται τὸ σῶμα ἔτι), οὔτε τὸ σῶμα μεῖζον ἢ ὁ τόπος· ἢ γὰρ κενὸν (205b.) ἔσται τι ἢ σῶμα οὐδαμοῦ πεφυκὸς εἶναι. Suppose (a) that the infinite sensible body is homogeneous. Then each part will be either immovable or always being carried along. Yet neither is possible. For why downwards rather than upwards or in any other direction? I mean, e.g, if you take a clod, where will it be moved or where will it be at rest? For ex hypothesi the place of the body akin to it is infinite. Will it occupy the whole place, then? And how? What then will be the nature of its rest and of its movement, or where will they be? It will either be at home everywhere—then it will not be moved; or it will be moved everywhere—then it will not come to rest. But if (b) the All has dissimilar parts, the proper places of the parts will be dissimilar also, and the body of the All will have no unity except that of contact. Then, further, the parts will be either finite or infinite in variety of kind. (i) Finite they cannot be, for if the All is to be infinite, some of them would have to be infinite, while the others were not, e.g. fire or water will be infinite. But, as we have seen before, such an element would destroy what is contrary to it. (This indeed is the reason why none of the physicists made fire or earth the one infinite body, but either water or air or what is intermediate between them, because the abode of each of the two was plainly determinate, while the others have an ambiguous place between up and down.) But (ii) if the parts are infinite in number and simple, their proper places too will be infinite in number, and the same will be true of the elements themselves. If that is impossible, and the places are finite, the whole too must be finite; for the place and the body cannot but fit each other. Neither is the whole place larger than what can be filled by the body (and then the body would no longer be infinite), nor is the body larger than the place; for either there would be an empty space or a body whose nature it is to be nowhere. Ἀναξαγόρας δ' ἀτόπως λέγει περὶ τῆς τοῦ ἀπείρου μονῆς· στηρίζειν γὰρ αὐτὸ αὑτό φησιν τὸ ἄπειρον· τοῦτο δέ, ὅτι ἐν αὑτῷ (ἄλλο γὰρ οὐδὲν περιέχειν), Anaxagoras gives an absurd account of why the infinite is at rest. He says that the infinite itself is the cause of its being fixed. This because it is in itself, since nothing else contains it ὡς ὅπου ἄν τι ᾖ, πεφυκὸς ἐνταῦθα εἶναι. τοῦτο δ' οὐκ ἀληθές· εἴη γὰρ ἄν τί που βιᾷ καὶ οὐχ οὗ πέφυκεν. εἰ οὖν ὅτι μάλιστα μὴ κινεῖται τὸ ὅλον (τὸ γὰρ αὑτῷ στηριζόμενον καὶ ἐν αὑτῷ ὂν ἀκίνητον εἶναι ἀνάγκη), ἀλλὰ διὰ τί οὐ πέφυκε κινεῖσθαι, λεκτέον. οὐ γὰρ ἱκανὸν τὸ οὕτως εἰπόντα ἀπηλλάχθαι· εἴη γὰρ ἂν καὶ ὅτι οὐκ ἔχει ἀλλαχῆ κινεῖσθαι οὐ κινούμενον, ἀλλὰ πεφυκέναι οὐδὲν κωλύει· ἐπεὶ καὶ ἡ γῆ οὐ φέρεται, οὐδ' εἰ ἄπειρος ἦν, εἰργμένη μέντοι ὑπὸ τοῦ μέσου· ἀλλ' οὐχ ὅτι οὐκ ἔστιν ἄλλο οὗ ἐνεχθήσεται, μείνειεν ἄν [ἐπὶ τοῦ μέσου], ἀλλ' ὅτι πέφυκεν οὕτω. καίτοι ἐξείη ἂν λέγειν ὅτι στηρίζει αὑτήν. εἰ οὖν μηδ' ἐπὶ τῆς γῆς τοῦτο αἴτιον ἀπείρου οὔσης, ἀλλ' ὅτι βάρος ἔχει, τὸ δὲ βαρὺ μένει ἐπὶ τοῦ μέσου, ἡ δὲ γῆ ἐπὶ τοῦ μέσου, ὁμοίως ἂν καὶ τὸ ἄπειρον μένοι ἐν αὑτῷ διά τιν' ἄλλην αἰτίαν, καὶ οὐχ ὅτι ἄπειρον καὶ στηρίζει αὐτὸ ἑαυτό. on the assumption that wherever anything is, it is there by its own nature. But this is not true: a thing could be somewhere by compulsion, and not where it is its nature to be. Even if it is true as true can be that the whole is not moved (for what is fixed by itself and is in itself must be immovable), yet we must explain why it is not its nature to be moved. It is not enough just to make this statement and then decamp. Anything else might be in a state of rest, but there is no reason why it should not be its nature to be moved. The earth is not carried along, and would not be carried along if it were infinite, provided it is held together by the centre. But it would not be because there was no other region in which it could be carried along that it would remain at the centre, but because this is its nature. Yet in this case also we may say that it fixes itself. If then in the case of the earth, supposed to be infinite, it is at rest, not because it is infinite, but because it has weight and what is heavy rests at the centre and the earth is at the centre, similarly the infinite also would rest in itself, not because it is infinite and fixes itself, but owing to some other cause. ἅμα δὲ δῆλον ὅτι κἂν ὁτιοῦν μέρος δέοι μένειν· ὡς γὰρ τὸ ἄπειρον ἐν ἑαυτῷ μένει στηρίζον, οὕτως κἂν ὁτιοῦν ληφθῇ μέρος ἐν ἑαυτῷ μενεῖ· τοῦ γὰρ ὅλου καὶ τοῦ μέρους ὁμοειδεῖς οἱ τόποι, οἷον ὅλης γῆς καὶ βώλου κάτω καὶ παντὸς πυρὸς καὶ σπινθῆρος ἄνω. ὥστε εἰ τοῦ ἀπείρου τόπος τὸ ἐν αὑτῷ, καὶ τοῦ μέρους ὁ αὐτός. μενεῖ ἄρα ἐν ἑαυτῷ. Another difficulty emerges at the same time. Any part of the infinite body ought to remain at rest. Just as the infinite remains at rest in itself because it fixes itself, so too any part of it you may take will remain in itself. The appropriate places of the whole and of the part are alike, e.g. of the whole earth and of a clod the appropriate place is the lower region; of fire as a whole and of a spark, the upper region. If, therefore, to be in itself is the place of the infinite, that also will be appropriate to the part. Therefore it will remain in itself. ὅλως δὲ φανερὸν ὅτι ἀδύνατον ἄπειρον ἅμα λέγειν σῶμα καὶ τόπον τινὰ εἶναι τοῖς σώμασιν, εἰ πᾶν σῶμα αἰσθητὸν ἢ βάρος ἔχει ἢ κουφότητα, καὶ εἰ μὲν βαρύ, ἐπὶ τὸ μέσον ἔχει τὴν φορὰν φύσει, εἰ δὲ κοῦφον, ἄνω· ἀνάγκη γὰρ καὶ τὸ ἄπειρον, ἀδύνατον δὲ ἢ ἅπαν ὁποτερονοῦν ἢ τὸ ἥμισυ ἑκάτερον πεπονθέναι· πῶς γὰρ διελεῖς; ἢ πῶς τοῦ ἀπείρου ἔσται τὸ μὲν ἄνω τὸ δὲ κάτω, ἢ ἔσχατον καὶ μέσον; In general, the view that there is an infinite body is plainly incompatible with the doctrine that there is necessarily a proper place for each kind of body, if every sensible body has either weight or lightness, and if a body has a natural locomotion towards the centre if it is heavy, and upwards if it is light. This would need to be true of the infinite also. But neither character can belong to it: it cannot be either as a whole, nor can it be half the one and half the other. For how should you divide it? or how can the infinite have the one part up and the other down, or an extremity and a centre? ἔτι πᾶν σῶμα αἰσθητὸν ἐν τόπῳ, τόπου δὲ εἴδη καὶ διαφοραὶ τἄνω καὶ κάτω καὶ ἔμπροσθεν καὶ ὄπισθεν καὶ δεξιὸν καὶ ἀριστερόν· καὶ ταῦτα οὐ μόνον πρὸς ἡμᾶς καὶ θέσει, ἀλλὰ καὶ ἐν αὐτῷ τῷ ὅλῳ διώρισται. ἀδύνατον δ' ἐν τῷ ἀπείρῳ εἶναι ταῦτα. Further, every sensible body is in place, and the kinds or differences of place are up-down, before-behind, right-left; and these distinctions hold not only in relation to us and by arbitrary agreement, but also in the whole itself. But in the infinite body they cannot exist. ἁπλῶς δ' εἰ ἀδύνατον (206a.) τόπον ἄπειρον εἶναι, ἐν τόπῳ δὲ πᾶν σῶμα, ἀδύνατον ἄπειρον [τι] εἶναι σῶμα. ἀλλὰ μὴν τό γε ποὺ ἐν τόπῳ, καὶ τὸ ἐν τόπῳ πού. εἰ οὖν μηδὲ ποσὸν οἷόν τ' εἶναι τὸ ἄπειρον—ποσὸν γὰρ τὶ ἔσται, οἷον δίπηχυ ἢ τρίπηχυ· ταῦτα γὰρ σημαίνει τὸ ποσόν—οὕτω καὶ τὸ ἐν τόπῳ ὅτι πού, τοῦτο δὲ ἢ ἄνω ἢ κάτω ἢ ἐν ἄλλῃ τινὶ διαστάσει τῶν ἕξ, τούτων δ' ἕκαστον πέρας τί ἐστιν. In general, if it is impossible that there should be an infinite place, and if every body is in place, there cannot be an infinite body. Surely what is in a special place is in place, and what is in place is in a special place. Just, then, as the infinite cannot be quantity—that would imply that it has a particular quantity, e,g, two or three cubits; quantity just means these—so a thing's being in place means that it is somewhere, and that is either up or down or in some other of the six differences of position: but each of these is a limit.
Postquam philosophus ostendit non esse corpus sensibile infinitum, facta suppositione quod sint elementa finita, hic ostendit idem simpliciter absque omni suppositione. 358. After showing that there is no infinite sensible body on the assumption that the elements are finite, the Philosopher here shows the same absolutely, without assumptions of any kind. Et primo dicit de quo est intentio; secundo exequitur propositum, ibi: aptum enim natum est et cetera. First he declares his intention; Secondly he carries out his proposal, at 359. Dicit ergo primo quod ex iis quae sequuntur oportet considerare de omni corpore universaliter, nulla suppositione facta, si contingat quodcumque corpus naturale esse infinitum. Et ex sequentibus rationibus manifestum fiet quod non. Deinde cum dicit: aptum enim natum est etc., ostendit propositum quatuor rationibus. Secunda incipit ibi: omnino autem manifestum etc.; tertia ibi: amplius omne corpus sensibile etc.; quarta ibi: simpliciter autem si impossibile et cetera. He says therefore first  that in what follows it is necessary to consider every body universally, without any suppositions, and ask whether any natural body can be infinite. And from the following reasons it will be clear that none can. Then, he proves his proposition with four reasons, beginning at 359. The second reason begins at 367; the third at 368; the fourth at 369. Circa primam rationem tria facit: primo praesupponit quaedam necessaria ad rationem; secundo ponit rationem ibi: quare si quidem sit eiusdem speciei etc.; tertio excludit quandam falsam opinionem, ibi: Anaxagoras autem inconvenienter et cetera. In regard to the first reason he does three things: First he lays down certain facts presupposed to his reasoning; Secondly he gives the reasoning itself, at 360. Thirdly, he excludes a false opinion, at 364. Praemittit ergo tria. Quorum primum est quod omne corpus sensibile habet aptitudinem naturalem ut sit in aliquo loco. Secundum est quod cuilibet corpori naturali convenit aliquis locus locorum qui sunt. Tertium est quod idem est locus naturalis totius et partis, sicut totius terrae et unius glebae, et totius ignis et unius scintillae: et huius signum est, quod in quacumque parte loci totius ponatur pars corporis, quiescit ibi. 359. Therefore  he lays down three presuppositions. The first of these is that every sensible body has a natural aptitude to be in some definite place. Secondly, that every natural body has, among available places, some place that befits it. Thirdly, that the natural place of the whole and of the part is the same, i.e., of all earth and each clod, of all fire and each spark. A sign of this is that in whatever part of the place of the whole there is placed a part of some body, it is at rest there. Deinde cum dicit: quare si quidem sit eiusdem speciei etc., ponit rationem, quae talis est. Si ponatur aliquod corpus infinitum, aut oportet quod totum sit unius speciei cum suis partibus, sicut aqua vel aer; aut quod habeat partes dissimilium specierum, ut homo aut planta. 360. Then [24171 he gives the [first] reason, which is this. If an infinite body be supposed, it must have parts either of the same species, as water or air, of parts of varying species, as a man or a plant has. Si habet omnes partes unius speciei, sequitur secundum praemissa, quod vel sit totaliter immobile et nunquam moveatur, aut quod semper moveatur. Quorum utrumque est impossibile: quia per alterum horum excluditur quies, et per alterum motus a rebus naturalibus, et utroque modo tollitur ratio naturae, cum natura sit principium motus et quietis. If all its parts are of the same species, it follows according to our pre-suppositions (no. 359) that it is either entirely immobile and is never moved, or is always being moved. But both of these are impossible: for in the second case, rest is excluded; and in the other, motion is excluded from natural things. Thus, in both cases there is denied the notion of nature, for nature is a principle of motion and of rest. Quod autem sequatur quod sit vel totaliter mobile vel totaliter quietum, probat consequenter per hoc, quod non esset assignare rationem quare aliquid magis sursum aut deorsum moveretur, aut in quamcumque partem. Et hoc manifestat per exemplum: ponamus enim quod totum illud corpus infinitum simile in partibus sit terra; non erit assignare ubi aliqua gleba terrae moveatur vel ubi quiescat; quia quamlibet partem loci infiniti occupabit aliquod corpus sibi cognatum, idest eiusdem speciei. Numquid igitur potest dici quod una gleba moveatur ad hoc quod contineat, idest quod occupet, successive totum locum infinitum, sicut sol movetur ut sit in qualibet parte circuli zodiaci? Et quomodo poterit hoc esse, ut una gleba terrae pertranseat per omnes partes infiniti loci? Nihil autem movetur ad impossibile: si igitur impossibile est quod gleba moveatur ad occupandum totum locum infinitum, ubi erit quies eius, et ubi motus eius? Aut enim oportet quod semper quiescat, et sic nunquam moveatur: aut quod semper moveatur, et sic nunquam quiescat. He proves that this body would be either entirely mobile or entirely at rest by the fact that no reason can be given for its being moved either up or down or in any direction whatsoever. He manifesto this by an example: for let us suppose that the entire infinite body which is similar throughout is earth. Then it will be impossible to say where any clod of earth should move or be at rest, because each part of infinite place would be occupied by some body related to it, i.e., of the same species. Can it be said that one clod of earth would be moved so as to occupy successively all the infinite places, as the sun is moved so as to be in each part of the zodiacal circle? And how could one clod of earth pass through all the parts of infinite place? Now nothing is moved toward the impossible: if therefore it is impossible for a clod of earth to be moved so as to occupy all the infinite places, in which place will it rest and in which will it be in motion? it will either always be at rest and thus never in motion or it will always be in motion and thus never at rest. Si autem detur alia pars divisionis, scilicet quod corpus infinitum habeat partes dissimiles secundum speciem; sequitur etiam quod dissimilia sint loca diversarum partium: alius est enim locus naturalis aquae, et alius terrae. Sed ex hac positione sequitur primo, quod corpus totius infiniti non sit unum simpliciter sed secundum quid, scilicet secundum contactum; et sic non erit unum corpus infinitum ut ponebatur. 361. If we suppose the other possibility, namely, that the infinite body has parts that are unlike in species, it will then follow that there would be unlike places for the unlike parts, for the natural place of water is one thing and that of earth is another, But on this supposition it follows at once that the body of this infinite whole would not be one body simply but one through contact; and thus there will not be one infinite body as our supposition granted. Et quia posset aliquis non reputare hoc inconveniens, subiungit aliam rationem contra hoc: et dicit quod si totum infinitum componitur ex dissimilibus partibus, necesse est quod huiusmodi partes dissimiles secundum speciem, aut sint specierum finitarum, aut infinitarum secundum numerum. Non autem potest esse quod sint finitarum specierum, quia oportebit, si totum est infinitum, quod quaedam sint finita secundum quantitatem, et quaedam infinita; aliter enim ex finitis numero posset componi infinitum: hoc autem posito, sequitur quod illa quae sunt infinita, corrumpant alia propter contrarietatem, ut prius dictum est in praecedenti ratione. Et ideo etiam nullus antiquorum naturalium philosophorum unum principium, quod dixit esse infinitum, posuit ignem vel terram, quae sunt extrema, sed magis aquam vel aerem vel aliquod medium, quia loca istorum erant manifesta et determinata, scilicet sursum et deorsum; non sic autem est de aliis, sed terra est deorsum respectu eorum, et ignis sursum. 362. Because someone might not consider this impossible [i.e., an infinite body of dissimilar parts], he adds another reason against this, saying that if the infinite whole is composed of unlike parts, these parts will be either of a finite number of species or of an infinite number. It will not be the first, because it then follows that if the whole is infinite, then some of the parts will be finite and some infinite; otherwise we would be able to get an infinite composed all of finites. On this assumption it follows that those which are infinite will corrupt the others, on account of contrariety, as was said in previous reasoning (nos. 354, 356). For this reason, no one of the early philosophers who posited one infinite principle, posited it to be fire or earth, which are extremes; rather they posited water or air or some medium between them, because the places of the former were evident, i.e., above and below, but it is not the same with the others, for earth is below in respect of them and fire above. Si vero aliquis accipiat aliam partem, scilicet quod corpora partialia sint infinita secundum speciem, sequitur quod etiam loca sint infinita secundum speciem, et quod elementa sint infinita. Si autem hoc est impossibile, quod elementa sint infinita, ut in primo probatum est, et quod loca etiam sint infinita, cum non sit possibile invenire infinitas species locorum; necesse est quod totum corpus sit finitum. 363. But if someone admits the other alternative, namely, that the parts of the body are infinite in species, it follows that their places also are infinite in species, and that the elements are infinite. But if it is impossible that the elements be infinite, as was already proved in Book I (l.11), and that places be infinite, since it is not possible to find infinite species of place, it is necessary to admit that the whole body is finite. Et quia concluserat ex infinitate corporum infinitatem locorum subiungit quod impossibile est non aequari corpus ad locum; quia non potest esse quod sit locus maior quam quantum contingit esse corpus, neque corpus potest esse infinitum si locus non est infinitus, et neque corpus potest esse maius quam locus quocumque modo. Quia si locus sit maior quam corpus, sequitur quod sit vacuum alicubi: aut si corpus sit maius quam locus, sequitur quod aliqua pars corporis non sit in aliquo loco. And because he had concluded to an infinity of places from the infinite of bodies, he adds that it is impossible not to equate body with place; for no place is greater than the body it contains, nor can a body be infinite if its place is not infinite, nor can in any way a body be greater than its place. This is so because if the place is greater than the body, there will be some empty place; if the body is greater than its place, then some part of the body is in no place. Deinde cum dicit: Anaxagoras autem etc., excludit quendam errorem. Et primo ponit ipsum: et dicit quod Anaxagoras dixit infinitum quiescere, sed inconvenienter assignavit rationem quietis eius. Dixit enim quod fulcit, idest sustentat, infinitum seipsum, quia est in se et non in alio, cum nihil ipsum contineat; et sic non possit extra se moveri. 364. Then  he excludes an error. First, he cites the error and says that Anaxagoras claimed that the infinite is at rest but gave an invalid reason for its rest. For he said that the infinite bears up, i.e., sustains itself since it exists in itself and not in something else, for nothing contains it. And thus it could not be moved outside itself. Secundo, ibi: tanquam ubi utique etc., improbat duabus rationibus quod dictum est. Quarum prima est quod Anaxagoras sic assignavit rationem de quiete infiniti, ac si ubi aliquid sit, ibi sit aptum natum esse: quia ex hac sola ratione dixit infinitum quiescere, quia est in seipso. Sed hoc non est verum quod ubi aliquid est, ibi semper aptum natum sit esse: quia aliquid est alicubi per violentiam, et non naturaliter. 365. Secondly [24R] he disproves this statement with two reasons. The first of which is that Anaxagoras so assigned his reason for the rest of the infinite as to suppose that wherever a thing is, that is its natural place, for the only reason he gave for saying that the infinite is at rest, is that it exists in itself. But it is not true that where a thing is, there it is always naturally disposed to be, because some things are somewhere by force and not by nature. Quamvis igitur hoc maxime verum sit, quod totum infinitum non movetur, quia sustentatur et manet in seipso, et sic est immobile: sed tamen dicendum erat quare non est aptum natum moveri. Non enim potest aliquis evadere sic, dicens quod non movetur infinitum: quia eadem ratione et de quolibet alio nihil prohibet quod non moveatur; sed sit aptum natum moveri. Quia et si terra esset infinita, sicut nunc non fertur quando est in medio, ita et tunc non ferretur quantum ad partem quae esset in medio: sed hoc non esset quia non haberet aliquid aliud ubi sustentaretur nisi in medio, sed quia non habet aptitudinem naturalem ut a medio moveatur. Si ergo ita est in terra, quod non est causa quare quiescat in medio, quia est infinita, sed quia gravitatem habet ex qua nata est manere in medio; similiter de quocumque alio infinito assignanda est causa quare quiescat; et non quia est infinitum, vel quia fulcit seipsum. Now although it is true that an infinite whole is not moved, because it is sustained and remains in itself and for that reason is immobile, yet a reason should be given why it is not naturally disposed to be moved. one cannot evade this simply by saying that the infinite is not moved, since by the same reasoning there is nothing to prevent any other body from not being moved while it might be naturally disposed to be moved. Because oven if earth were infinite, just as now it will not be carried further when it is in the center, so even then no part in the center would move further: but this would not be because it had no other natural place except the center where it could be sustained but because it does not have a natural aptitude to be moved from the center. If, therefore, this is the case with earth, that the reason why it rests at the center is not that it is infinite but that it has gravity which accounts for its remaining in the center; similarly, in the case of any other infinite, the reason why it rests should be given, and this is not simply because it is infinite or that it supports itself. Aliam autem rationem ponit ibi: similiter autem manifestum et cetera. Et dicit quod si totum infinitum quiescit quia manet in seipso, sequitur quod quaelibet pars ex necessitate quiescat quia manet in seipsa. Idem enim est locus totius et partis, ut dictum est, ut ignis et scintillae sursum, et terrae et glebae deorsum. Si ergo totius infiniti locus est ipsummet, sequitur quod quaelibet pars infiniti maneat in seipsa sicut in proprio loco. 366. He lays down another argument . Thus he states that if the whole infinite is in repose because it remains within itself, it follows that any part thereof necessarily is necessarily at rest since it remains within itself. For the place of the whole and the part is the same, as was said (no. 359), e.g., that of fire and a spark upwards, that of earth and a clod of earth downward. If, therefore, the place of the whole infinite is itself, it follows that any part of the infinite will remain at rest within itself as in its proper place. Secundam rationem ponit ibi: omnino autem manifestum est et cetera. Et dicit quod omnino manifestum est quod impossibile est dicere esse infinitum corpus in actu, et quod cuiuslibet corporis est aliquis locus, si omne corpus sensibile aut habet gravitatem aut levitatem, sicut antiqui dixerunt ponentes infinitum. Quia si sit corpus grave, oportet quod naturaliter feratur ad medium: si autem sit leve, necesse est quod feratur sursum. Si ergo sit aliquod infinitum corpus sensibile, necesse est quod etiam in corpore infinito sit sursum et medium: sed impossibile est quod totum infinitum sustineat in se utrumlibet horum, scilicet vel sursum vel medium; vel etiam quod sustineat utrumque secundum diversas medietates. Quomodo enim infinitum poterit dividi, ut una pars eius sit sursum et alia deorsum, vel quod in infinito sit ultimum aut medium? Non est igitur corpus sensibile infinitum. 367. He gives a second reason against Anaxagoras  saying that it is clearly impossible to say that there is an actually infinite body and that there is some place for each body, if every sensible body is either heavy or light, as the ancients said who posited the infinite. Because if the body is heavy, it will be naturally carried to the center; if it is light, it will be carried upward. If, therefore, there be an infinite sensible body, there must be in it an “up” and a center. But it is impossible that the infinite body should sustain in itself either of these, i.e., either an “up” or a center, or even that it sustain both according to different centers. For how could the infinite be divided so that one part would be “up” and another “down” or how can there be in the infinite a boundary or a center? Therefore there is no infinite sensible body. Tertiam rationem ponit ibi: amplius omne corpus sensibile in loco est et cetera. Et dicit quod omne corpus sensibile est in loco. Differentiae autem loci sunt sex: sursum, deorsum, ante et retro, dextrorsum et sinistrorsum; quae quidem sunt determinata non solum quoad nos, sed etiam in ipso toto universo. 368. He gives the third reason  saying that every sensible body is in place. But the differences of place are six: above, and below, before and behind, to the right and to the left—and these are determined not only in relation to us but even in the whole universe itself. Determinantur enim secundum se huiusmodi positiones, in quibus sunt determinata principia et termini motus. Unde in animatis determinantur sursum et deorsum secundum motum alimenti; ante et retro secundum motum sensus; dextrorsum et sinistrorsum secundum motum processivum, cuius principium est a parte dextra. In rebus autem inanimatis, in quibus non sunt principia determinata horum motuum, dicitur dextrorsum et sinistrorsum per comparationem ad nos: dicitur enim columna dextra, quae est ad dextram hominis, et sinistra quae est ad sinistram. For such positions are determined in themselves in those things in which there are determinate principles and terms of motion. Whence in living things “up” and “down” are determined according to the movement of food; “front” and “rear” according to the movement of sense; “right” and “left” according to forward motion, which begins from the right. But in inanimate things, in which there are no determinate principles of such motions, “right” and “left” are said with respect to us—for a column is said to be “at the right” which is to the right of a man, and “at the left” which is at his left. Sed in toto universo determinatur sursum et deorsum secundum motum gravium et levium: secundum autem motum caeli determinatur dextrum oriens, sinistrum occidens; ante vero hemisphaerium superius, retro vero hemisphaerium inferius; sursum vero meridies, deorsum vero Septentrio. Haec autem non possunt determinari in corpore infinito: impossibile est ergo totum universum esse infinitum. But in the whole universe “up” and “down” are determined according to the movement of heavy and light things; while according to the motion of the heavens the rising sun determines “right,” the setting sun, “left”; “front” is determined by the upper hemisphere, “rear” by the lower hemisphere; “above” by the south, “below” by the north. Now such things cannot be determined in an infinite body. It is therefore impossible for the whole universe to be infinite. Quartam rationem ponit ibi: simpliciter autem si impossibile est et cetera. Et dicit quod si impossibile est esse locum infinitum, cum omne corpus sit in loco, sequitur quod impossibile sit esse aliquod corpus infinitum. Sed quod impossibile sit esse locum infinitum, sic probat: quia haec duo convertuntur, esse in loco et esse in aliquo loco; sicut et esse hominem et esse aliquem hominem, et esse quantitatem et esse aliquam quantitatem. Sicut igitur impossibile est esse quantitatem infinitam, quia sequeretur aliquam quantitatem esse infinitam, ut bicubitum et tricubitum, quod est impossibile; ita impossibile est esse locum infinitum, quia sequeretur aliquem locum infinitum esse, vel sursum vel deorsum et huiusmodi: quod est impossibile, cum quodlibet eorum significet quendam terminum, ut dictum est. Sic igitur nullum corpus sensibile est infinitum. 369. Then  he gives the fourth reason, saying that if it is impossible that there be an infinite place because every body is in a place, it follows that there can be no infinite body. That an infinite place is impossible be proves thus: To be in place and to be in some place are convertible, just as to be man and to be some man or to be quantity and to be some quantity. Therefore, just as it is impossible that there be infinite quantity, because then it would follow that some quantity is infinite, e.g., two cubits or three cubits, which is impossible, so infinite place is impossible, because it would follow that some place is infinite (either up or down or some other place), which is impossible—since each of these implies a definite term as was said (in 368). Therefore no sensible body is infinite.
The infinite as existing in potency
Chapter 5 cont. ὅτι μὲν οὖν ἐνεργείᾳ οὐκ ἔστι σῶμα ἄπειρον, φανερὸν ἐκ τούτων. It is plain from these arguments that there is no body which is actually infinite. Chapter 6 Ὅτι δ' εἰ μὴ ἔστιν ἄπειρον ἁπλῶς, πολλὰ ἀδύνατα συμβαίνει, δῆλον. τοῦ τε γὰρ χρόνου ἔσται τις ἀρχὴ καὶ τελευτή, καὶ τὰ μεγέθη οὐ διαιρετὰ εἰς μεγέθη, καὶ ἀριθμὸς οὐκ ἔσται ἄπειρος. ὅταν δὲ διωρισμένων οὕτως μηδετέρως φαίνηται ἐνδέχεσθαι, διαιτητοῦ δεῖ, καὶ δῆλον ὅτι πὼς μὲν ἔστιν πὼς δ' οὔ. But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and an end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. If, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in; and clearly there is a sense in which the infinite exists and another in which it does not. λέγεται δὴ τὸ εἶναι τὸ μὲν δυνάμει τὸ δὲ ἐντελεχείᾳ, καὶ τὸ ἄπειρον ἔστι μὲν προσθέσει ἔστι δὲ καὶ διαι ρέσει. τὸ δὲ μέγεθος ὅτι μὲν κατ' ἐνέργειαν οὐκ ἔστιν ἄπειρον, εἴρηται, διαιρέσει δ' ἐστίν· οὐ γὰρ χαλεπὸν ἀνελεῖν τὰς ἀτόμους γραμμάς· λείπεται οὖν δυνάμει εἶναι τὸ ἄπειρον. We must keep in mind that the word 'is' means either what potentially is or what fully is. Further, a thing is infinite either by addition or by division. Now, as we have seen, magnitude is not actually infinite. But by division it is infinite. (There is no difficulty in refuting the theory of indivisible lines.) The alternative then remains that the infinite has a potential existence. οὐ δεῖ δὲ τὸ δυνάμει ὂν λαμβάνειν, ὥσπερ εἰ δυνατὸν τοῦτ' ἀνδριάντα εἶναι, ὡς καὶ ἔσται τοῦτ' ἀνδριάς, οὕτω καὶ ἄπειρον ὃ ἔσται ἐνεργείᾳ· ἀλλ' ἐπεὶ πολλαχῶς τὸ εἶναι, ὥσπερ ἡ ἡμέρα ἔστι καὶ ὁ ἀγὼν τῷ ἀεὶ ἄλλο καὶ ἄλλο γίγνεσθαι, οὕτω καὶ τὸ ἄπειρον (καὶ γὰρ ἐπὶ τούτων ἔστι καὶ δυνάμει καὶ ἐνεργείᾳ· Ὀλύμπια γὰρ ἔστι καὶ τῷ δύνασθαι τὸν ἀγῶνα γίγνεσθαι καὶ τῷ γίγνεσθαι)· But the phrase 'potential existence' is ambiguous. When we speak of the potential existence of a statue we mean that there will be an actual statue. It is not so with the infinite. There will not be an actual infinite. The word 'is' has many senses, and we say that the infinite 'is' in the sense in which we say 'it is day' or 'it is the games', because one thing after another is always coming into existence. For of these things too the distinction between potential and actual existence holds. We say that there are Olympic games, both in the sense that they may occur and that they are actually occurring. ἄλλως δ' ἔν τε τῷ χρόνῳ δῆλον [τὸ ἄπειρον] καὶ ἐπὶ τῶν ἀνθρώπων, καὶ ἐπὶ τῆς διαιρέσεως τῶν μεγεθῶν. The infinite exhibits itself in different ways—in time, in the generations of man, and in the division of magnitudes. ὅλως μὲν γὰρ οὕτως ἔστιν τὸ ἄπειρον, τῷ ἀεὶ ἄλλο καὶ ἄλλο λαμβάνεσθαι, καὶ τὸ λαμβανόμενον μὲν ἀεὶ εἶναι πεπερασμένον, ἀλλ' ἀεί γε ἕτερον καὶ ἕτερον· [ἔτι τὸ εἶναι πλεοναχῶς λέγεται, ὥστε τὸ ἄπειρον οὐ δεῖ λαμβάνειν ὡς τόδε τι, οἷον ἄνθρωπον ἢ οἰκίαν, ἀλλ' ὡς ἡ ἡμέρα λέγεται καὶ ὁ ἀγών, οἷς τὸ εἶναι οὐχ ὡς οὐσία τις γέγονεν, ἀλλ' ἀεὶ ἐν γενέσει ἢ φθορᾷ, πεπερασμένον, ἀλλ' ἀεί γε ἕτερον καὶ ἕτερον·] For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different. Again, 'being' has more than one sense, so that we must not regard the infinite as a 'this', such as a man or a horse, but must suppose it to exist in the sense in which we speak of the day or the games as existing things whose being has not come to them like that of a substance, but consists in a process of coming to be or passing away; definite if you like at each stage, yet always different. ἀλλ' ἐν (206b.) τοῖς μεγέθεσιν ὑπομένοντος τοῦ ληφθέντος [τοῦτο συμβαίνει], ἐπὶ δὲ τοῦ χρόνου καὶ τῶν ἀνθρώπων φθειρομένων οὕτως ὥστε μὴ ἐπιλείπειν. But when this takes place in spatial magnitudes, what is taken perists, while in the succession of time and of men it takes place by the passing away of these in such a way that the source of supply never gives out. τὸ δὲ κατὰ πρόσθεσιν τὸ αὐτό ἐστί πως καὶ τὸ κατὰ διαίρεσιν· ἐν γὰρ τῷ πεπερασμένῳ κατὰ πρόσθεσιν γίγνεται ἀντεστραμμένως· ᾗ γὰρ διαιρούμενον ὁρᾶται εἰς ἄπειρον, ταύτῃ προστιθέμενον φανεῖται πρὸς τὸ ὡρισμένον. ἐν γὰρ τῷ πεπερασμένῳ μεγέθει ἂν λαβών τις ὡρισμένον προσλαμβάνῃ τῷ αὐτῷ λόγῳ, μὴ τὸ αὐτό τι τοῦ ὅλου μέγεθος περιλαμβάνων, οὐ διέξεισι τὸ πεπερασμένον· ἐὰν δ' οὕτως αὔξῃ τὸν λόγον ὥστε ἀεί τι τὸ αὐτὸ περιλαμβάνειν μέγεθος, διέξεισι, διὰ τὸ πᾶν πεπερασμένον ἀναιρεῖσθαι ὁτῳοῦν ὡρισμένῳ. ἄλλως μὲν οὖν οὐκ ἔστιν, οὕτως δ' ἔστι τὸ ἄπειρον, δυνάμει τε καὶ ἐπὶ καθαιρέσει (καὶ ἐντελεχείᾳ δὲ ἔστιν, ὡς τὴν ἡμέραν εἶναι λέγομεν καὶ τὸν ἀγῶνα)· καὶ δυνάμει οὕτως ὡς ἡ ὕλη, καὶ οὐ καθ' αὑτό, ὡς τὸ πεπερασμένον. καὶ κατὰ πρόσθεσιν δὴ οὕτως ἄπειρον δυνάμει ἔστιν, ὃ ταὐτὸ λέγομεν τρόπον τινὰ εἶναι τῷ κατὰ διαίρεσιν· ἀεὶ μὲν γάρ τι ἔξω ἔσται λαμβάνειν, In a way the infinite by addition is the same thing as the infinite by division. In a finite magnitude, the infinite by addition comes about in a way inverse to that of the other. For in proportion as we see division going on, in the same proportion we see addition being made to what is already marked off. For if we take a determinate part of a finite magnitude and add another part determined by the same ratio (not taking in the same amount of the original whole), and so on, we shall not traverse the given magnitude. But if we increase the ratio of the part, so as always to take in the same amount, we shall traverse the magnitude, for every finite magnitude is exhausted by means of any determinate quantity however small. The infinite, then, exists in no other way, but in this way it does exist, potentially and by reduction. It exists fully in the sense in which we say 'it is day' or 'it is the games'; and potentially as matter exists, not independently as what is finite does. By addition then, also, there is potentially an infinite, namely, what we have described as being in a sense the same as the infinite in respect of division. For it will always be possible to take something ab extra. οὐ μέντοι ὑπερβαλεῖ παντὸς μεγέθους, ὥσπερ ἐπὶ τὴν διαίρεσιν ὑπερβάλλει παντὸς ὡρισμένου καὶ ἀεὶ ἔσται ἔλαττον. Yet the sum of the parts taken will not exceed every determinate magnitude, just as in the direction of division every determinate magnitude is surpassed in smallness and there will be a smaller part. ὥστε δὲ παντὸς ὑπερβάλλειν κατὰ τὴν πρόσθεσιν, οὐδὲ δυνάμει οἷόν τε εἶναι, εἴπερ μὴ ἔστι κατὰ συμβεβηκὸς ἐντελεχείᾳ ἄπειρον, ὥσπερ φασὶν οἱ φυσιολόγοι τὸ ἔξω σῶμα τοῦ κόσμου, οὗ ἡ οὐσία ἢ ἀὴρ ἢ ἄλλο τι τοιοῦτον, ἄπειρον εἶναι. ἀλλ' εἰ μὴ οἷόν τε εἶναι ἄπειρον ἐντελεχείᾳ σῶμα αἰσθητὸν οὕτω, φανερὸν ὅτι οὐδὲ δυνάμει ἂν εἴη κατὰ πρόσθεσιν, ἀλλ' ἢ ὥσπερ εἴρηται ἀντεστραμμένως τῇ διαιρέσει, But in respect of addition there cannot be an infinite which even potentially exceeds every assignable magnitude, unless it has the attribute of being actually infinite, as the physicists hold to be true of the body which is outside the world, whose essential nature is air or something of the kind. But if there cannot be in this way a sensible body which is infinite in the full sense, evidently there can no more be a body which is potentially infinite in respect of addition, except as the inverse of the infinite by division, as we have said. ἐπεὶ καὶ Πλάτων διὰ τοῦτο δύο τὰ ἄπειρα ἐποίησεν, ὅτι καὶ ἐπὶ τὴν αὔξην δοκεῖ ὑπερβάλλειν καὶ εἰς ἄπειρον ἰέναι καὶ ἐπὶ τὴν καθαίρεσιν. ποιήσας μέντοι δύο οὐ χρῆται· οὔτε γὰρ ἐν τοῖς ἀριθμοῖς τὸ ἐπὶ τὴν καθαίρεσιν ἄπειρον ὑπάρχει (ἡ γὰρ μονὰς ἐλάχιστον), οὔτε <�τὸ> ἐπὶ τὴν αὔξην (μέχρι γὰρ δεκάδος ποιεῖ τὸν ἀριθμόν). It is for this reason that Plato also made the infinites two in number, because it is supposed to be possible to exceed all limits and to proceed ad infinitum in the direction both of increase and of reduction. Yet though he makes the infinites two, he does not use them. For in the numbers the infinite in the direction of reduction is not present, as the monad is the smallest; nor is the infinite in the direction of increase, for the parts number only up to the decad.
Postquam philosophus disputative processit de infinito, hic incipit determinare veritatem. 370. After discussing the infinite dialectically, the Philosopher now begins to determine the truth. Et primo ostendit an sit infinitum; secundo quid sit, ibi: accidit autem contrarium et cetera. First he determines whether there is an infinite; Secondly, what it is, at 382. Prima dividitur in duas: in prima ostendit quomodo infinitum sit; in secunda comparat diversa infinita ad invicem, ibi: aliter autem et in tempore et cetera. The first is divided into two parts: In the first, he shows how the infinite exists; In the second, he compares various infinites one to the other, at 374. Circa primum tria facit: primo ostendit quod infinitum quodammodo est, et quodammodo non est; secundo determinat quod est in potentia, et non est sicut actu ens, ibi: dicitur igitur etc.; tertio manifestat quomodo sit in potentia, ibi: non oportet autem potentia ens et cetera. About the first he does three things: First, he shows that the infinite in a way exists and in a way it does not; Secondly, he shows that it is in potency and is not as a being in act, at 372; Thirdly, he manifests how it is in potency, at 373. Dicit ergo primo quod ex praemissis manifestum est, quod non sit aliquod corpus infinitum in actu. Item ex iis quae ante dicta sunt, manifestum est quod si infinitum simpliciter non sit, quod multa impossibilia accidunt. Quorum unum est quod tempus habebit principium et finem: quod reputatur inconveniens secundum ponentes aeternitatem mundi. Et iterum sequetur quod magnitudo non semper sit divisibilis in magnitudines, sed quandoque deveniatur per divisionem magnitudinum ad quaedam quae non sunt magnitudines: sed omnis magnitudo est divisibilis. Item sequetur quod numerus non augeatur in infinitum. Quia igitur secundum determinata neutrum videtur contingere, neque scilicet quod infinitum sit actu, neque quod simpliciter non sit; necesse est dicere quod quodammodo est, quodammodo non est. 371. Accordingly, he says first  that from the foregoing (ll.8.9) it is manifest that there is no infinite body in act. It is also clear from what has been said (l.7) that if the infinite absolutely does not exist, many impossibilities arise. One is that time will have a beginning and an end, considered impossible by those holding for the eternity of the world. Another is that it would follow that a magnitude would not be always divisible into further magnitudes, but eventually one would arrive through division of magnitudes at certain things which are not magnitudes. But every magnitude is divisible. Likewise, it would follow that number could not increase to infinity. Since therefore, according to what has been said (ll.7-9) neither seems to occur, i.e.,) either an infinite in act or no infinite at all, it must be said that the infinite somehow is and somehow is not. Deinde cum dicit: dicitur igitur esse aliud etc., ostendit quod infinitum est sicut potentia ens. Et dicit quod aliquid dicitur esse in actu, et aliquid dicitur esse in potentia. Infinitum autem dicitur esse per appositionem, sicut in numeris, vel per ablationem, sicut in magnitudinibus. Ostensum est enim quod magnitudo non est actu infinita; et sic in magnitudinibus per appositionem infinitum non invenitur, sed per divisionem in eis invenitur infinitum. Non enim est difficile destruere opinionem ponentium indivisibiles esse lineas. Vel, secundum aliam litteram: non est difficile partiri atomos lineas, idest ostendere lineas, quas quidam ponunt indivisibiles, esse partibiles. Dicitur autem infinitum in appositione vel divisione, secundum quod potest apponi vel dividi. Relinquitur igitur quod infinitum sit tanquam in potentia ens. 372. Then  he shows that the infinite is as a being in potency. And he says that something is said to be in act and something is said to be in potency. Now the infinite is said to come about either by addition, as in numbers, or by subtraction, as in magnitudes. Now it has been shown that magnitude is not infinite in act; hence in magnitudes an infinite through addition is not found, but there is found in them an infinite through division. For it is easy to destroy the opinion that posits lines as indivisibles, or according to another letter, it is easy “to divide indivisible lines,” i.e., to show that lines held indivisible by some, are divisible. Now the infinite, whether in addition or division, is spoken of to the extent of the ability [or potency] to add or divide. It therefore follows that the infinite is as a being in potency. Deinde cum dicit: non oportet autem potentia ens etc., ostendit quomodo infinitum sit in potentia. Dupliciter enim invenitur aliquid in potentia. Uno modo sic quod totum potest reduci in actum, sicut possibile est hoc aes esse statuam, quod aliquando erit statua; non autem sic dicitur esse infinitum in potentia, quod postea totum sit in actu. Alio modo aliquid dicitur in potentia esse, quod postea fit actu ens, non quidem totum simul, sed successive. Multipliciter enim dicitur aliquid esse: vel quia totum est simul, ut homo et domus; vel quia semper una pars eius fit post aliam, per quem modum dicitur esse dies et ludus agonalis. 373. Then  he shows how the infinite exists in potency. For something is found to be in potency in two ways. In one way, in the sense that the whole can be reduced to act, as it is possible for this bronze to be a statue, because at some time it will be a statue. But the infinite in potency is not so meant as that which later will be entirely in act. In another way, something is said to be in potency in such a way that later it will be in act, not, indeed, all at once, but part after part. For there are many ways in which a thing is said to be: 1) because the whole exists at the same time, as in the case of a man or a house; or 2) because one part of it always comes to be after another part, in the way that a day is said to exist and a competition exists. Et hoc modo dicitur infinitum esse simul et in potentia et in actu: omnia enim huiusmodi simul sunt in potentia quantum ad unam partem, et in actu quantum ad aliam. Olympia enim, idest festa agonalia quae celebrabantur in monte Olympo, dicuntur esse et durare secundum agones posse fieri et fieri in actu: quia quamdiu durabant ista festa, aliqua pars illorum ludorum erat in fieri, et aliqua erat ut in futurum fienda. It is in this latter way that the infinite is said to be at once in potency and in act. For all successive things are at once in potency as to one part and in act as to another part. For the Olympic games, i.e., the festive contests held on Mt. Olympus, are said to be and to continue as long as the contests are scheduled and as long as the schedule is being carried out. For as long as those games lasted, one part of the schedule was taking place at the time and another was to take place later. Deinde cum dicit: aliter autem et in tempore etc., comparat diversa infinita ad invicem. 374. Then  he compares various infinites one to another: Et primo comparat infinitum temporis et generationis, infinito quod est in magnitudinibus; secundo comparat infinitum secundum appositionem et infinitum secundum divisionem in magnitudinibus, ibi: quod autem secundum appositionem et cetera. First he compares the infinite of time and of generation to the infinite which is in magnitudes; Secondly, he compares the infinite according to addition to the infinite according to division in the case of magnitudes, at 377. Circa primum tria facit. Primo proponit quod intendit: et dicit quod aliter manifestatur infinitum in generatione hominum et in tempore, et aliter in divisione magnitudinum. In regard to the first he does three things: First, he proposes his intention and says that the infinite in the generation of man, and in time, must be explained in a manner different from that of the infinite in the division of magnitudes. Secundo ibi: omnino quidem enim sic est etc., ostendit quid sit commune omnibus infinitis. Et dicit quod hoc omnino et universaliter in omnibus infinitis invenitur, quod infinitum est in semper aliud et aliud accipiendo secundum quandam successionem, ita tamen quod quidquid accipitur in actu de infinito, totum sit finitum. Unde non oportet accipere quod infinitum sit aliquid totum simul existens, sicut hoc aliquid demonstratum, sicut accipimus hominem vel domum; sed sicut sunt successiva, ut dies et ludus agonalis, quorum esse non est hoc modo quod aliquid eorum sit sicut quaedam substantia perfecta tota actu existens. In generatione autem et corruptione, etsi in infinitum procedatur, semper illud quod accipitur in actu, est finitum. In toto enim decursu generationis, etiam si procedatur in infinitum, et omnes homines qui simul actu accipiuntur, sunt finiti secundum numerum, et huiusmodi finitum oportet accipere alterum et alterum, secundum quod quidam homines succedunt quibusdam. 375. Secondly  he shows what is common to all infinites, saying in all of them it is universally found that the infinite consists in always taking one thing followed by another according to some certain succession, in such away that the whole of whatever is taken, be finite. Hence one must not suppose the infinite to be some whole existing all at once, as a substance that can be pointed out, e.g., a man or a house. Rather the infinite must be taken as in the case of successive things, such as a day or a tournament, whose existence is not that of a perfect substance actually existing as a complete whole all at once. Now, in generation and corruption, even though the process continue to infinity, whatever is taken in act is finite. For in the whole course of generation, even should it proceed to infinity, both all the men existing at a given time are finite in number, and this finite amount must be taken as other and other, accordingly as men succeed one another in time. Tertio ibi: sed in magnitudinibus etc., ostendit differentiam. Et dicit quod illud finitum quod accipimus in magnitudinibus, vel apponendo vel dividendo, permanet et non corrumpitur: sed illa finita quae accipiuntur in infinito decursu temporis et generationis humanae corrumpuntur; ita quod per istum modum non contingat tempus et generationem deficere. 376. Thirdly,  he shows how they differ, saying that the finite actually present in magnitudes as a result of adding or of dividing is permanent and is not corrupted, but the finites considered in the infinite course of time and of human generation are corrupted, although in such a way that time and generation themselves do not fail. Deinde cum dicit: quod autem secundum appositionem etc., comparat duo infinita quae sunt in magnitudinibus, scilicet secundum appositionem et secundum divisionem. Et circa hoc tria facit: primo ponit convenientiam inter utrumque infinitum; secundo ostendit differentiam, ibi: non tamen excellit etc.; tertio infert quandam conclusionem ex dictis, ibi: quare excellere et cetera. 377. Then  he compares the two types of infinite which are found in magnitudes; namely, the infinite according to addition and the infinite according to division. About this he does three things: First he shows their points of agreement; Secondly, he shows wherein they differ, at 379; Thirdly, he draws a conclusion from what has been said, at 380. Dicit ergo primo quod quodammodo infinitum secundum appositionem est idem cum infinito secundum divisionem; quia infinitum secundum appositionem fit e converso cum infinito secundum divisionem. Secundum enim quod aliquid dividitur in infinitum, secundum hoc in infinitum videtur posse apponi ad aliquam determinatam quantitatem. In regard to the first,  he says that in some sense the infinite resulting from addition is the same as the one resulting from division, because the former comes to be as a converse of the latter. For it is accordingly as something is divided to infinity, that additions to infinity seem to be able to be made to some determinate quantity. Manifestat igitur quomodo sit infinitum divisione in magnitudine. 378. He demonstrates, therefore, how the infinite in division exists in magnitude. Et dicit quod si aliquis in aliqua magnitudine finita, accepta aliqua parte determinata per divisionem, semper accipiat dividendo alias partes secundum eandem rationem, idest proportionem, sed non secundum eandem quantitatem in eadem proportione, non pertransibit dividendo illud finitum; Thus he states that if someone, in some infinite magnitude, having taken some determinate part by division, should then continue to take other parts by division, always maintaining the same ratio, I.e., proportion, he will not go through that finite magnitude by means of division. puta si a linea cubitali accipiat medietatem, et iterum a residuo medietatem; et sic in infinitum procedere potest. Servabitur enim in subtrahendo eadem proportio, sed non eadem quantitas subtracti; minus est enim secundum quantitatem dimidium dimidii quam dimidium totius. Sed si semper sumeret eandem quantitatem, oporteret quod semper magis ac magis augeretur proportio. Puta si a quantitate decem cubitorum subtrahatur unus cubitus, subtractum se habet ad totum in subdecupla proportione: si autem iterum a residuo subtrahatur unus cubitus, subtractum se habebit in maiori proportione; minus enim unus cubitus exceditur a novem quam a decem. Sicut igitur servando eandem proportionem diminuitur quantitas, ita sumendo eandem quantitatem augetur proportio. Si ergo aliquis sic subtrahendo ab aliqua magnitudine finita, semper augeat proportionem sumendo eandem quantitatem, transibit dividendo magnitudinem finitam; puta si a linea centum cubitorum semper subtrahat unum cubitum. Et hoc ideo est, quia omne finitum consumitur quocumque finito semper accepto. For example, from a line of one cubit we may take one half, and from the remainder one-half again. We can proceed in this process to infinity. For the same proportion will be maintained in subtracting, but not the same amount of what is subtracted. The half of the half is less, according to quantity, than the half of the whole. But if we were to take away always the same amount, the proportion taken away would be continually growing. For example, if from a quantity of ten cubits, we take away one cubit, the ratio of the part removed to the original is one-tenth. If we take from the remainder another inch, the ratio between the part removed and that which remains will be in a greater proportion [i.e., one-ninth]. For one cubit is less exceeded by 9 than by 10. Just as, by preserving the same proportion throughout, the quantity subtracted is continually smaller, so, by taking away the same amount each time, the proportion gets continually larger. If, therefore, by so subtracting from some finite magnitude, we continually increase the proportion by taking away the same amount, the original magnitude will be exhausted. For example, if from a line of 10 cubits we always subtract one cubit. This will happen because every finite thing will be exhausted by continually removing the same finite amount. Aliter igitur infinitum non est secundum divisionem, nisi in potentia, quod tamen simul est actu cum potentia, sicut dictum est de die et de agone. Et cum infinitum sit semper in potentia, assimilatur materiae, quae est semper in potentia; et non est per se existens in actu totum, sicut finitum est in actu. Et sicut infinitum secundum divisionem est in potentia cum actu simul, similiter dicendum est de infinito secundum appositionem, quod quodammodo est idem cum infinito secundum divisionem, ut dictum est. Inde autem manifestum est quod infinitum per appositionem est in potentia, quia semper contingit aliquid aliud accipere apponendo. The infinite that depends on division does not exist, therefore, except in potency, but with this potency there exists always something in act, as was said of a day or of a tournament. And since the infinite is always in potency, it is assimilated to matter, which likewise is always in potency; and it never exists in act in its entirety, the way that the finite is in act. And just as the infinite according to division is at once in potency and act, so too is the infinite according to addition, which has been shown to be in some sense the same as the infinite according to division, as was said (no. 377). And the reason why the infinite according to addition is in potency is that it can always grow through addition. Deinde cum dicit: non tamen excellit etc., ostendit differentiam inter infinitum secundum appositionem et infinitum secundum divisionem. Et dicit quod infinitum per appositionem non excedit in maius omnem magnitudinem finitam datam; sed infinitum secundum divisionem excedit omnem determinatam parvitatem in minus. Accipiamus enim aliquam determinatam parvitatem, puta unius digiti: si lineam centum cubitorum dividam in infinitum, accipiendo semper dimidium, venietur ad aliquid minus uno digito. Sed apponendo in infinitum, e contrario divisioni, erit dare aliquam quantitatem finitam quae nunquam pertransibitur. Dentur enim duae magnitudines, quarum utraque sit decem cubitorum, et tertia quae sit viginti. Si igitur id quod subtraho in infinitum, accipiendo semper dimidium ab una magnitudine decem cubitorum, addatur alteri quae etiam est decem cubitorum, nunquam pervenietur in infinitum apponendo ad mensuram quantitatis quae est viginti cubitorum: quia quantum remanebit in magnitudine cui subtrahitur, tantum deficiet a data mensura in quantitate cui addetur. 379. Then  he shows the difference between the infinite according to addition and the infinite according to division. And he says that the former does not exceed any given finite magnitude, whereas the latter diminishes beyond any pre-determined smallness. For If we take any predetermined smallness, for example, the width of a finger, we can, by repeated halving of a line of 10 cubits, arrive at a remainder which is less than the width of a finger. But in adding to infinity, in distinction to division, there will exist some given finite quantity which will never be gone through. Take two magnitudes each of 10 cubits, and a third one of 20 cubits. If what I subtract to infinity from one magnitude of 10 cubits, always taking a half, is added to the other, which is also of 10 cubits, I shall never reach, by adding to infinity, the measure of the quantity of 20 cubits, since as much as remains in the quantity being divided will be lacking from the given measure in the quantity being added to. Deinde cum dicit: quare excellere omne etc., inducit conclusionem ex dictis. Et primo inducit eam; secundo manifestat per dictum Platonis, ibi: quoniam et Plato et cetera. 380. Then  he draws a conclusion from the foregoing. First he draws it; secondly he explains it by a saying of Plato, at 381. Dicit ergo primo quod ex quo appositio in infinitum non facit transcendere omnem determinatam quantitatem, non est possibile esse, nec etiam in potentia, quod excellatur omnis determinata quantitas per appositionem. Quia si esset in natura potentia ad appositionem transcendentem omnem quantitatem, sequeretur quod esset actu infinitum; sic quod infinitum esset accidens alicui naturae, sicut naturales philosophi extra corpus huius mundi quod videmus, ponunt quod est quoddam infinitum, cuius substantia est aer vel aliquid aliud huiusmodi. Si ergo non est possibile esse corpus sensibile actu infinitum, ut ostensum est, sequitur quod non sit potentia in natura ad appositionem transcendentem omnem magnitudinem; sed solum ad appositionem infinitam per contrarium divisioni, ut dictum est. He says therefore first  that since addition to infinity never actually transcends every determined quantity, it is not possible, even in potency, to transcend every determined quantity by addition. For if there were in nature potency for addition transcending every quantity, it would follow that something actually infinite exists; such an infinite would be an accident of some nature, in the same way that the natural philosophers posit, outside the world we see, some sort of infinite, whose substance is air or something similar. But if, as was shown (ll.8,9), no infinite sensible body exists in act, it follows that there is in nature no potency to transcend every magnitude by addition, but only a potency to the infinite addition which is in contrast to [and derived from] division, as was said above (no. 379). Quare autem si esset potentia ad infinitam additionem transcendentem omnem magnitudinem, sequatur esse corpus infinitum in actu, non autem ad additionem infinitam in numeris, transcendentem omnem numerum, sequatur esse numerum infinitum in actu, infra ostendetur. Why the existence of a potency to infinite addition transcending every magnitude would imply a body infinite in act, whereas in numbers infinite addition transcending every number does not imply an actually infinite umber will be explained below(in Lecture 12). Deinde cum dicit: quoniam et Plato propter hoc etc., manifestat quod dixerat per dictum Platonis. Et dicit quod quia infinitum in appositione magnitudinum est per oppositum divisioni, propter hoc Plato duo fecit infinita, scilicet magnum, quod pertinet ad additionem, et parvum quod pertinet ad divisionem; quia scilicet infinitum videtur excellere et per additionem in augmentum, et per divisionem in decrementum, vel tendendo in nihil. Sed cum ipse Plato faciat duo infinita, non tamen utitur eis: quia cum numerum poneret substantiam esse omnium rerum, in numeris non invenitur infinitum per divisionem, quia in eis est minimum unitas; neque etiam per additionem secundum ipsum, quia dicebat quod species numerorum non variantur nisi usque ad decem, et postea reditur ad unitatem, computando undecim et duodecim et cetera. 381. Then  he confirms what he has said by a dictum of Plato, saying that because the infinite resulting from the addition of magnitudes is the reverse of division, Plato therefore posited two infinites: “the large,” which pertains to addition, and “the small,” which pertains to division—for the finite seems to excel both by addition unto increase, and by division unto decrease or towards nothing. Yet although Plato makes two infinites, he does not use them. For in number, which he posited to be the substance of all things, there is no infinite by division since there is among them something smallest, which is unity; nor is there according to him an infinite according by addition, since he said that the species of number vary only up to ten and then a return is made to unity when we count eleven, twelve, and so on.
Definition of the infinite
Chapter 6 cont. συμβαίνει δὲ τοὐναντίον εἶναι ἄπειρον ἢ ὡς λέγουσιν. (207a.) οὐ γὰρ οὗ μηδὲν ἔξω, ἀλλ' οὗ ἀεί τι ἔξω ἐστί, τοῦτο ἄπειρόν ἐστιν. The infinite turns out to be the contrary of what it is said to be. It is not what has nothing outside it that is infinite, but what always has something outside it. σημεῖον δέ· καὶ γὰρ τοὺς δακτυλίους ἀπείρους λέγουσι τοὺς μὴ ἔχοντας σφενδόνην, ὅτι αἰεί τι ἔξω ἔστι λαμβάνειν, καθ' ὁμοιότητα μέν τινα λέγοντες, οὐ μέντοι κυρίως· δεῖ γὰρ τοῦτό τε ὑπάρχειν καὶ μηδέ ποτε τὸ αὐτὸ λαμβά νεσθαι· ἐν δὲ τῷ κύκλῳ οὐ γίγνεται οὕτως, ἀλλ' αἰεὶ τὸ ἐφεξῆς μόνον ἕτερον. ἄπειρον μὲν οὖν ἐστιν οὗ κατὰ τὸ ποσὸν λαμβάνουσιν αἰεί τι λαμβάνειν ἔστιν ἔξω. This is indicated by the fact that rings also that have no bezel are described as 'endless', because it is always possible to take a part which is outside a given part. The description depends on a certain similarity, but it is not true in the full sense of the word. This condition alone is not sufficient: it is necessary also that the next part which is taken should never be the same. In the circle, the latter condition is not satisfied: it is only the adjacent part from which the new part is different. Our definition then is as follows: A quantity is infinite if it is such that we can always take a part outside what has been already taken. οὗ δὲ μηδὲν ἔξω, τοῦτ' ἔστι τέλειον καὶ ὅλον· οὕτω γὰρ ὁριζόμεθα τὸ ὅλον, οὗ μηδὲν ἄπεστιν, οἷον ἄνθρωπον ὅλον ἢ κιβώτιον. ὥσπερ δὲ τὸ καθ' ἕκαστον, οὕτω καὶ τὸ κυρίως, οἷον τὸ ὅλον οὗ μηδέν ἐστιν ἔξω· οὗ δ' ἔστιν ἀπουσία ἔξω, οὐ πᾶν, ὅ τι ἂν ἀπῇ. ὅλον δὲ καὶ τέλειον ἢ τὸ αὐτὸ πάμπαν ἢ σύνεγγυς τὴν φύσιν. τέλειον δ' οὐδὲν μὴ ἔχον τέλος· τὸ δὲ τέλος πέρας. On the other hand, what has nothing outside it is complete and whole. For thus we define the whole—that from which nothing is wanting, as a whole man or a whole box. What is true of each particular is true of the whole as such—the whole is that of which nothing is outside. On the other hand that from which something is absent and outside, however small that may be, is not 'all'. 'Whole' and 'complete' are either quite identical or closely akin. Nothing is complete (teleion) which has no end (telos); and the end is a limit. διὸ βέλτιον οἰητέον Παρμενίδην Μελίσσου εἰρηκέναι· ὁ μὲν γὰρ τὸ ἄπειρον ὅλον φησίν, ὁ δὲ τὸ ὅλον πεπεράνθαι, "μεσσόθεν ἰσοπαλές". οὐ γὰρ λίνον λίνῳ συνάπτειν ἐστὶν τῷ ἅπαντι καὶ ὅλῳ τὸ ἄπειρον, Hence Parmenides must be thought to have spoken better than Melissus. The latter says that the whole is infinite, but the former describes it as limited, 'equally balanced from the middle'. For to connect the infinite with the all and the whole is not like joining two pieces of string; ἐπεὶ ἐντεῦθέν γε λαμβάνουσι τὴν σεμνότητα κατὰ τοῦ ἀπείρου, τὸ πάντα περιέχειν καὶ τὸ πᾶν ἐν ἑαυτῷ ἔχειν, διὰ τὸ ἔχειν τινὰ ὁμοιότητα τῷ ὅλῳ. ἔστι γὰρ τὸ ἄπειρον τῆς τοῦ μεγέθους τελειότητος ὕλη καὶ τὸ δυνάμει ὅλον, ἐντελεχείᾳ δ' οὔ, διαιρετὸν δ' ἐπί τε τὴν καθαίρεσιν καὶ τὴν ἀντεστραμμένην πρόσθεσιν, ὅλον δὲ καὶ πεπερασμένον οὐ καθ' αὑτὸ ἀλλὰ κατ' ἄλλο· καὶ οὐ περιέχει ἀλλὰ περιέχεται, ᾗ ἄπειρον. διὸ καὶ ἄγνωστον ᾗ ἄπειρον· εἶδος γὰρ οὐκ ἔχει ἡ ὕλη. ὥστε φανερὸν ὅτι μᾶλλον ἐν μορίου λόγῳ τὸ ἄπειρον ἢ ἐν ὅλου· μόριον γὰρ ἡ ὕλη τοῦ ὅλου ὥσπερ ὁ χαλκὸς τοῦ χαλκοῦ ἀνδριάντος, for it is from this they get the dignity they ascribe to the infinite—its containing all things and holding the all in itself—from its having a certain similarity to the whole. It is in fact the matter of the completeness which belongs to size, and what is potentially a whole, though not in the full sense. It is divisible both in the direction of reduction and of the inverse addition. It is a whole and limited; not, however, in virtue of its own nature, but in virtue of what is other than it. It does not contain, but, in so far as it is infinite, is contained. Consequently, also, it is unknowable, qua infinite; for the matter has no form. (Hence it is plain that the infinite stands in the relation of part rather than of whole. For the matter is part of the whole, as the bronze is of the bronze statue.) ἐπεὶ εἴ γε περιέχει ἐν τοῖς αἰσθητοῖς, καὶ ἐν τοῖς νοητοῖς τὸ μέγα καὶ τὸ μικρὸν ἔδει περιέχειν τὰ νοητά. ἄτοπον δὲ καὶ ἀδύνατον τὸ ἄγνωστον καὶ ἀόριστον περιέχειν καὶ ὁρίζειν. If it contains in the case of sensible things, in the case of intelligible things the great and the small ought to contain them. But it is absurd and impossible to suppose that the unknowable and indeterminate should contain and determine.
Postquam philosophus ostendit quomodo est infinitum, hic ostendit quid sit infinitum. Et circa hoc tria facit: primo ostendit quid sit infinitum; secundo ex hoc assignat rationem eorum quae de infinito dicuntur, ibi: secundum rationem autem accidit etc.; tertio solvit rationes quae supra positae sunt, ibi: reliquum autem est et cetera. 382. After showing how the infinite exists, the Philosopher now explains what it is. About this he does three things: First he shows what the infinite is; Secondly, from this he assigns the reason for the things said of the infinite, at 390 (l.12). Thirdly, he solves the difficulties mentioned earlier, at 400 (l.13). Circa primum duo facit: primo ostendit quid sit infinitum, excludens quorundam falsam definitionem; secundo excludit quandam falsam opinionem consequentem ex illa falsa definitione, ibi: quoniam hinc accipiunt dignitatem et cetera. In regard to the first he does two things: First he shows what the infinite is and rejects the false definition of some; Secondly, he rejects a certain false opinion that follows from the above false definition, at 387. Circa primum tria facit: primo proponit quod intendit; secundo manifestat propositum, ibi: signum autem etc.; tertio infert quandam conclusionem ex dictis, ibi: unde melius est opinandum et cetera. About the first he does three things: First, he proposes what he intends; Secondly, he explains the proposition, at 384; Thirdly, he draws a conclusion, at 386. Dicit ergo primo quod contrario modo definiendum est infinitum quam sicut quidam definierunt. Dixerunt enim quidam quod infinitum est extra quod nihil est: sed e contra dicendum est quod infinitum est cuius est semper aliquid extra. 383. He says therefore  that the infinite must be defined in a manner contrary to the way some have defined it. For some have said that infinite is ‘that outside of which there is nothing,” whereas, to the contrary, it should be defined as “that beyond which there is always something.” Deinde cum dicit: signum autem est etc., manifestat propositum. 384. Then  he explains his proposition. Et primo ostendit quod sua assignatio sit bona; secundo quod assignatio antiquorum sit incompetens, ibi: cuius autem nihil est extra et cetera. First he shows that his description is good; Secondly, that the description of the earlier philosophers is incompetent, at 385. Ostendit ergo primo quod infinitum sit cuius semper est aliquid extra, per quoddam signum. Dicunt enim quidam quod annuli sunt infiniti, quia per hoc quod habent quandam circulationem, semper est ibi supponere partem ad partem acceptam. Sed hoc dicitur secundum similitudinem, et non proprie: quia ad hoc quod aliquid sit infinitum, requiritur hoc, scilicet quod extra quamlibet partem acceptam sit quaedam alia; ita tamen quod nunquam resumatur illa quae prius fuit accepta. Sed in circulo non est sic, quia pars quae accipitur post aliam partem, est alia solum ab ea quae immediate accepta est, non tamen ab omnibus partibus prius acceptis; quia una pars potest multoties sumi, ut patet in motu circulari. Si igitur annuli dicuntur infiniti propter hanc similitudinem, sequitur quod illud quod est vere infinitum, sit cuius semper sit accipere aliquid extra, si aliquis velit accipere eius quantitatem. Non enim potest comprehendi quantitas infiniti; sed si quis velit eam accipere, accipiet partem post partem in infinitum, ut supra dictum est. He shows therefore first by an example that the infinite is “that beyond which there is always something.” For some people say that a ring is infinite since, because it has a circular direction, one can always take part after part. But this is to speak analogously and not properly, because to be infinite requires this, namely, that beyond whatever part is taken there be some other part, in such a way, nevertheless, that one never take again a part taken previously. But in a circle this does not happen, because the part which is counted after another happens to be different from that immediately before it, but not from all the parts previously counted, because one part can be counted any number of times, as is evident in a circular motion. Therefore, if rings are called infinite according to this analogy, it follows that that which is truly infinite is something which always has something beyond, if one were to measure its quantity. For it is impossible to measure the quantity of the infinite; but if someone should desire to reckon it, he would take part after part to infinity, as said above. Deinde cum dicit: cuius autem nihil est etc., probat quod definitio antiquorum sit incompetens, tali ratione. Id cuius nihil est extra est definitio perfecti et totius. Quod sic probat. Definitur enim unumquodque totum esse cui nihil deest: sicut dicimus hominem totum aut arcam totam, quibus nihil deest eorum quae debent habere. Et sicut hoc dicimus in aliquo singulari toto, ut est hoc particulare vel illud, ita etiam haec ratio competit in eo quod est vere et proprie totum, scilicet in universo, extra quod simpliciter nihil est. Cum autem aliquid desit per absentiam alicuius intrinseci, tunc non est totum. 385. Then  he proves that the definition of the earlier philosophers is since “that outside of which there is nothing” is a definition of the perfect and a whole thing. Here is his proof. Every whole is defined as “that to which nothing to lacking”—as we speak of a whole man, or of a whole box, if they lack nothing which they ought to have. And just as we speak thus in regard to some individual whole, as in the case of this or that particular, so too this notion holds in regard to what is truly and perfectly whole, namely, that outside of which there is absolutely nothing. But when something is lacking through the absence of something intrinsic, then such a thing is not a whole. Sic igitur manifestum est quod haec est definitio totius: totum est cuius nihil est extra. Sed totum et perfectum vel sunt penitus idem, vel sunt propinqua secundum naturam. Et hoc ideo dicit, quia totum non invenitur in simplicibus, quae non habent partes: in quibus tamen utimur nomine perfecti. Per hoc igitur manifestum est quod perfectum est cuius nihil est extra ipsum. Sed nullum carens fine est perfectum; quia finis est perfectio uniuscuiusque. Finis autem est terminus eius cuius est finis: nullum igitur infinitum et interminatum est perfectum. Non ergo competit infinito definitio perfecti, cuius scilicet nihil est extra. So it is evident that this is the definition of a whole: “a whole is that nothing of which is outside of it.” But a whole thing and a perfect thing are either entirely the same or of a proximate nature. He says this, because “whole” is not found in simple things which have no parts; in which things, nevertheless, we use the word “perfect.” This shows that the perfect is “that which has nothing of itself outside of it.” But nothing that lacks an end is perfect, because the end is the perfection of each thing. For the end is the term of that of which it is the end. Nothing infinite, therefore, and unterminated, is perfect. Hence the definition of the perfect as that, namely, which has nothing of itself outside itself, does not apply to the infinite. Deinde cum dicit: unde melius est opinandum etc., inducit quandam conclusionem ex dictis. Quia enim infinito non competit definitio totius, manifestum est quod melius dixit Parmenides quam Melissus. Melissus enim dixit totum universum esse infinitum; Parmenides vero dixit quod totum finitur per aeque pugnans a medio, in quo designavit corpus universi esse sphaericum. In sphaerica enim figura lineae a medio usque ad terminum, scilicet circumferentiam, ducuntur secundum aequalitatem, quasi aeque pugnantes sibi invicem. Et recte dicitur quod totum universum sit finitum, quia totum et infinitum non se invicem consequuntur quasi sibi continuata, sicut lino continuatur linum in filando. Erat enim proverbium, ut ea quae se consequuntur, dicerentur sibi continuari sicut linum lino. 386. Then  he draws a conclusion from the foregoing. Since the definition of a “whole” does not apply to the infinite, it is clear that the position of Parmenides is better than that of Melissus. For Melissus said that the whole universe was infinite. Parmenides said the whole is terminated by what is “striving equally from the middle,” by which he designated the body of the universe as spherical. For in a spherical figure, lines from the center to the term, i.e., the circumference, are drawn according to equality, as though “striving equally” with each other. And it is rightly stated that the whole universe is finite, for to be a whole and to be infinite are not reciprocally connected, i.e., not continuous as thread follows thread in spinning. For there was a proverb that things which follow one upon the other should be said to be continuous as thread following thread. Deinde cum dicit: quoniam hinc accipiunt etc., excludit quandam falsam opinionem ex praedicta definitione falsa exortam. Et primo communiter quantum ad omnes; secundo specialiter quantum ad Platonem, ibi: quoniam si continet et cetera. 387. Then  he rejects a false opinion that arose from the aforesaid definition, and first in a general way, covering all variations; secondly the opinion of Plato, at 389. Dicit ergo primo quod quia aestimaverunt infinitum coniungi toti, hinc acceperunt quasi dignitatem, idest rem per se notam, de infinito, quod omnia contineret et omnia in se haberet; propter hoc quod habet quandam similitudinem cum toto, sicut id quod est in potentia habet similitudinem cum actu. Infinitum enim inquantum est in potentia, est sicut materia respectu perfectionis magnitudinis: et est sicut totum in potentia, non autem in actu. Quod patet ex hoc, quod infinitum dicitur secundum quod possibile est aliquid dividi in minus, et secundum quod ex opposito divisioni potest fieri appositio, ut supra dictum est. Sic igitur infinitum secundum se, idest secundum propriam rationem, est in potentia totum: et est imperfectum, sicut materia non habens perfectionem. He says therefore first that because some thought that whole and infinite were mutually connected, they consequently took it as a “dignity” [axium], i.e., something self-evident, that the infinite contains all things and that it has all things in itself. This was due to the fact that the infinite has a likeness to a whole, as what is in potency has a likeness to act. For the infinite, inasmuch as it is in potency, is as matter in respect to the perfection of magnitude, and it is as a whole in potency, not as a whole in act. This is proved by the fact that the infinite is based on the possibility of dividing things into what is smaller and of making, by a contrasting division, continual additions, as was said above (l.10). Consequently, the infinite in itself, according to its proper nature, is a whole in potency only; and it is something imperfect, comparable to matter not having perfection. Non autem est totum et finitum secundum se, idest secundum propriam rationem qua est infinitum; sed secundum aliud, idest secundum finem et totum, ad quod est in potentia. Divisio enim quae est possibilis in infinitum, secundum quod ad aliquid terminatur, dicitur esse perfecta: et secundum quod vadit in infinitum, est imperfecta. Et manifestum est quod, cum totius sit continere, materiae autem contineri, quod infinitum inquantum huiusmodi non continet, sed continetur: inquantum scilicet id quod de infinito est in actu, semper continetur ab aliquo maiori, secundum quod possibile est aliquid extra accipere. For it is not whole and infinite [or finished] according to itself, i.e., according to proper notion by which it is infinite, but according to something other, i.e., according to end and whole, to which it is in potency. For division, which is possible ad infinitum, is called “perfect” insofar as it is, whereas the division that goes on ad infinitum is imperfect. And it is clear, since it is the whole that contains but matter that is contained, that the infinite an such does not contain but is contained. This is true, insofar, namely, as whatever is in act of the infinite is always contained by something greater, accordingly as it is possible to take something beyond. Ex hoc autem quod est sicut ens in potentia, non solum hoc sequitur, quod infinitum contineatur et non contineat: sed etiam sequuntur duae aliae conclusiones. Quarum una est, quod infinitum inquantum huiusmodi est ignotum, quia est sicut materia non habens speciem, idest formam, ut dictum est; materia autem non cognoscitur nisi per formam. Alia conclusio est, quae ex eodem sequitur, quod infinitum magis habet rationem partis quam totius, quia materia comparatur ad totum ut pars. Et recte infinitum se habet ut pars, inquantum non est de ipso accipere nisi aliquam partem in actu. 388. Now from the fact that the infinite is as a being in potency, not only does it follow that the infinite is contained and does not contain, but two other conclusions also follow. One is that the infinite, as such, is unknown, because it is as matter without species, i.e., form, and matter is not known except through form. The other conclusion, which has the same source, is that the infinite has more the notion of a part than that of a whole, since matter is compared to the whole as a part. And it is not a surprise that the infinite conducts itself as a part, inasmuch as only a part of it is ever actual. Deinde cum dicit: quoniam si continet etc., excludit opinionem Platonis, qui ponebat infinitum tam in sensibilibus quam in intelligibilibus. Et dicit quod ex hoc manifestum est etiam quod, si magnum et parvum, quibus Plato attribuit infinitum, sunt in sensibilibus et in intelligibilibus tanquam continentia (propter hoc quod continere attribuitur infinito); sequitur quod infinitum contineat intelligibilia. Sed hoc videtur esse inconveniens et impossibile, quod infinitum, cum sit ignotum et indeterminatum, contineat et determinet intelligibilia. Non enim determinantur nota per ignota, sed magis e converso. 389. Then  he rejects an opinion of Plato who posited an infinite both in sensible and in intelligible things. And he states that from this it is plain also that if “the large” and “the small.” to which Plato attributed infinity, are in sensible and intelligible things as containing (by virtue of containment being attributed to the infinite), it follows that the infinite contains the intelligible things. But this seems unfitting and impossible, namely, that the infinite, since it is unknown and undetermined, should contain and determine intelligible things. For the known is not determined by the unknown, but rather the converse is true.
Explanations in the light of the definition of the infinite
Chapter 7 Κατὰ λόγον δὲ συμβαίνει καὶ τὸ κατὰ πρόσθεσιν μὲν μὴ εἶναι δοκεῖν ἄπειρον οὕτως ὥστε παντὸς ὑπερβάλλειν μεγέθους, ἐπὶ τὴν διαίρεσιν δὲ εἶναι (περιέχεται γὰρ ἡ ὕλη (207b.) ἐντὸς καὶ τὸ ἄπειρον, περιέχει δὲ τὸ εἶδος)· It is reasonable that there should not be held to be an infinite in respect of addition such as to surpass every magnitude, but that there should be thought to be such an infinite in the direction of division. For the matter and the infinite are contained inside what contains them, while it is the form which contains. εὐλόγως δὲ καὶ τὸ ἐν μὲν τῷ ἀριθμῷ εἶναι ἐπὶ μὲν τὸ ἐλάχιστον πέρας ἐπὶ δὲ τὸ πλεῖον ἀεὶ παντὸς ὑπερβάλλειν πλήθους, ἐπὶ δὲ τῶν μεγεθῶν τοὐναντίον ἐπὶ μὲν τὸ ἔλαττον παντὸς ὑπερβάλλειν μεγέθους ἐπὶ δὲ τὸ μεῖζον μὴ εἶναι μέγεθος ἄπειρον. αἴτιον δ' ὅτι τὸ ἕν ἐστιν ἀδιαίρετον, ὅ τι περ ἂν ἓν ᾖ (οἷον ἄνθρωπος εἷς ἄνθρωπος καὶ οὐ πολλοί), ὁ δ' ἀριθμός ἐστιν ἕνα πλείω καὶ πόσ' ἄττα, ὥστ' ἀνάγκη στῆναι ἐπὶ τὸ ἀδιαίρετον (τὸ γὰρ τρία καὶ δύο παρώνυμα ὀνόματά ἐστιν, ὁμοίως δὲ καὶ τῶν ἄλλων ἀριθμῶν ἕκαστος), It is natural too to suppose that in number there is a limit in the direction of the minimum, and that in the other direction every assigned number is surpassed. In magnitude, on the contrary, every assigned magnitude is surpassed in the direction of smallness, while in the other direction there is no infinite magnitude. The reason is that what is one is indivisible whatever it may be, e.g. a man is one man, not many. Number on the other hand is a plurality of 'ones' and a certain quantity of them. Hence number must stop at the indivisible: for 'two' and 'three' are merely derivative terms, and so with each of the other numbers. ἐπὶ δὲ τὸ πλεῖον ἀεὶ ἔστι νοῆσαι· ἄπειροι γὰρ αἱ διχοτομίαι τοῦ μεγέθους. ὥστε δυνάμει μὲν ἔστιν, ἐνεργείᾳ δ' οὔ· ἀλλ' ἀεὶ ὑπερβάλλει τὸ λαμβανόμενον παντὸς ὡρισμένου πλήθους. ἀλλ' οὐ χωριστὸς ὁ ἀριθμὸς οὗτος [τῆς διχοτομίας], οὐδὲ μένει ἡ ἀπειρία ἀλλὰ γίγνεται, ὥσπερ καὶ ὁ χρόνος καὶ ὁ ἀριθμὸς τοῦ χρόνου. But in the direction of largeness it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence this infinite is potential, never actual: the number of parts that can be taken always surpasses any assigned number. But this number is not separable from the process of bisection, and its infinity is not a permanent actuality but consists in a process of coming to be, like time and the number of time. ἐπὶ δὲ τῶν μεγεθῶν τοὐναντίον ἐστί· διαιρεῖται μὲν γὰρ εἰς ἄπειρα τὸ συνεχές, ἐπὶ δὲ τὸ μεῖζον οὐκ ἔστιν ἄπειρον. ὅσον γὰρ ἐνδέχεται δυνάμει εἶναι, καὶ ἐνεργείᾳ ἐνδέχεται τοσοῦτον εἶναι. ὥστε ἐπεὶ ἄπειρον οὐδέν ἐστι μέγεθος αἰσθητόν, οὐκ ἐνδέχεται παντὸς ὑπερβολὴν εἶναι ὡρισμένου μεγέθους· εἴη γὰρ ἄν τι τοῦ οὐρανοῦ μεῖζον. With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no infinite in the direction of increase. For the size which it can potentially be, it can also actually be. Hence since no sensible magnitude is infinite, it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens. τὸ δ' ἄπειρον οὐ ταὐτὸν ἐν μεγέθει καὶ κινήσει καὶ χρόνῳ, ὡς μία τις φύσις, ἀλλὰ τὸ ὕστερον λέγεται κατὰ τὸ πρότερον, οἷον κίνησις μὲν ὅτι τὸ μέγεθος ἐφ' οὗ κινεῖται ἢ ἀλλοιοῦται ἢ αὐξάνεται, ὁ χρόνος δὲ διὰ τὴν κίνησιν. νῦν μὲν οὖν χρώμεθα τούτοις, ὕστερον δὲ ἐροῦμεν καὶ τί ἐστιν ἕκαστον, καὶ διότι πᾶν μέγεθος εἰς μεγέθη διαιρετόν. The infinite is not the same in magnitude and movement and time, in the sense of a single nature, but its secondary sense depends on its primary sense, i.e. movement is called infinite in virtue of the magnitude covered by the movement (or alteration or growth), and time because of the movement. (I use these terms for the moment. Later I shall explain what each of them means, and also why every magnitude is divisible into magnitudes.) οὐκ ἀφαιρεῖται δ' ὁ λόγος οὐδὲ τοὺς μαθηματικοὺς τὴν θεωρίαν, ἀναιρῶν οὕτως εἶναι ἄπειρον ὥστε ἐνεργείᾳ εἶναι ἐπὶ τὴν αὔξησιν ἀδιεξίτητον· οὐδὲ γὰρ νῦν δέονται τοῦ ἀπείρου (οὐ γὰρ χρῶνται), ἀλλὰ μόνον εἶναι ὅσην ἂν βούλωνται πεπερασμένην· τῷ δὲ μεγίστῳ μεγέθει τὸν αὐτὸν ἔστι τετμῆσθαι λόγον ὁπηλικονοῦν μέγεθος ἕτερον. ὥστε πρὸς μὲν τὸ δεῖξαι ἐκείνοις οὐδὲν διοίσει τὸ [δ'] εἶναι ἐν τοῖς οὖσιν μεγέθεσιν. Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable. In point of fact they do not need the infinite and do not use it. They postulate only that the finite straight line may be produced as far as they wish. It is possible to have divided in the same ratio as the largest quantity another magnitude of any size you like. Hence, for the purposes of proof, it will make no difference to them to have such an infinite instead, while its existence will be in the sphere of real magnitudes. ἐπεὶ δὲ τὰ αἴτια διῄρηται τετραχῶς, φανερὸν ὅτι ὡς ὕλη τὸ ἄπειρον αἴτιόν ἐστι, καὶ ὅτι (208a.) τὸ μὲν εἶναι αὐτῷ στέρησις, τὸ δὲ καθ' αὑτὸ ὑποκείμενον τὸ συνεχὲς καὶ αἰσθητόν. φαίνονται δὲ πάντες καὶ οἱ ἄλλοι ὡς ὕλῃ χρώμενοι τῷ ἀπείρῳ· διὸ καὶ ἄτοπον τὸ περιέχον ποιεῖν αὐτὸ ἀλλὰ μὴ περιεχόμενον. In the fourfold scheme of causes, it is plain that the infinite is a cause in the sense of matter, and that its essence is privation, the subject as such being what is continuous and sensible. All the other thinkers, too, evidently treat the infinite as matter—that is why it is inconsistent in them to make it what contains, and not what is contained.
Posita definitione infiniti, hic ex definitione assignata assignat rationem eorum quae de infinito dicuntur. 390. After giving a definition of the infinite, the Philosopher now assigns reasons for the things that are said about the infinite. Et primo eius quod dicitur de appositione et divisione infiniti; secundo eius quod infinitum in diversis secundum ordinem invenitur, ibi: infinitum autem non idem est etc.; tertio eius quod mathematici utuntur infinito, ibi: non removet autem ratio etc.; quarto eius quod infinitum ponitur principium, ibi: quoniam autem causae et cetera. First, the reason for what is said about addition and division of the infinite; Secondly, the reason for saying that the infinite is found in different things according to a certain order, at 397; Thirdly, the reason for saying that mathematicians use the infinite, at 398; Fourthly, the reason why the infinite is called a principle, at 399. Circa primum duo facit: primo assignat rationem eius quod dicitur de infinito, circa divisionem vel appositionem in magnitudinibus; secundo eius quod dicitur in numeris, per comparationem ad magnitudines, ibi: rationabiliter autem est et cetera. About the first he does two things: First he presents the reason for what is said about the infinite in relation to division and addition in magnitudes; Secondly, the reason for what is said of it in numbers by comparison to magnitudes, at 392. Dictum est autem supra quod appositio in infinitum sic invenitur in magnitudinibus, quod tamen non exceditur per hoc quaecumque determinata magnitudo. Sed divisio in infinitum sic invenitur in magnitudinibus, quod dividendo transitur quaecumque quantitas in minus, ut supra expositum est. Hoc autem secundum rationem dicit accidere: quia cum infinitum habeat rationem materiae, continetur intus sicut materia: illud autem quod continet, est species et forma. Manifestum est autem ex iis quae dicta sunt in secundo, quod totum habet rationem formae, partes autem rationem materiae. Cum ergo in magnitudinibus a toto itur ad partes per divisionem, rationabile est quod ibi nullus terminus inveniatur, qui non transcendatur per infinitam divisionem. Sed in additione itur a partibus ad totum, quod habet rationem formae continentis et terminantis: unde rationabile est quod sit aliqua determinata quantitas, quam infinita appositio non transcendat. 391. It was said above (no. 379) that addition to infinity in magnitudes takes place in such a way that the resulting magnitude does not become greater than any given magnitude. But division to infinity in magnitudes results in reaching a quantity that is smaller than any pre-assigned quantity, as was expounded above (no. 379). However, he states  that this occurs reasonably, for since the infinite is like matter it is contained within just as matter is, while that which contains is the species and form. Now it is clear from what was said in Book II (l.5) that the whole is like form and the parts are like matter. Since, therefore, the division of a magnitude proceeds from the whole to the parts, it is reasonable that no limit be found there which is not transcended through infinite division. But the process of addition goes from the parts to the whole, which is like a form that contains and terminates; hence it is reasonable that there be some definite quantity which infinite addition does not exceed. Deinde cum dicit: rationabiliter autem est etc., assignat rationem de infinito in numeris, per comparationem ad magnitudines. Dicitur enim quod in numero invenitur aliquis terminus in minus, quem non est dividendo transcendere: sed non invenitur aliquis terminus in plus; quia quolibet numero est invenire alium maiorem per additionem. In magnitudinibus autem est e converso, ut dictum est. Et huius rationem assignat; et primo quidem quare in numeris aliquis terminus invenitur, qui in minus non transcenditur dividendo. Huius autem ratio est, quia omne unum, inquantum unum, est indivisibile, sicut homo indivisibilis est unus homo et non multi. Quemlibet autem numerum oportet resolvere in unum: quod patet ex ipsa ratione numeri. Numerus enim hoc significat, quod sint aliqua plura uno: quaelibet autem plura excedentia unum plus vel minus, sunt determinatae species numerorum. Unde cum unum sit de ratione numeri, et de ratione unius sit indivisibilitas, sequitur quod divisio numeri stet in termino indivisibili. 392. Then  he explains infinity in numbers by comparison to magnitudes. For it was said that in number there is a smallest terminus below which division does not go; but there is no maximum limit which it cannot exceed, because it is possible through addition to exceed any given number. The opposite however takes place in magnitudes, as was said (no. 391). The reason why is because every unity, inasmuch as it is a unity, is indivisible, as indivisible man is one man and not many men. Now every number can be resolved into unity, as is evident from the nature of number. For number signifies that there are more things than one, and any plurality exceeding one to a greater or lesser degree constitutes a definite species of number. Hence, since unity pertains to the notion of number and indivisibility pertains to the notion of unity, It follows that the division of number should halt at an indivisible terminus. Quod autem dixerat, quod de ratione numeri sit quod sint plura uno, manifestat per species; quia duo et tria et quilibet alius numerus denominatur ab uno. Unde dicitur in V Metaphys. quod substantia senarii est in hoc quod sit sexies unum, non autem in hoc quod sit bis tria vel ter duo: quia sequeretur quod unius rei essent plures definitiones et plures substantiae; quia ex diversis partibus diversimode consurgit unus numerus. This statement that it is of the nature of number to be more than unity he explains by appealing to the species of number, because 2 and 3 and every other number is denominated by unity. Wherefore it is said in Metaphysics that the substance of 6 consists in its being six times one and not two times three, or three times two. Otherwise, it would follow that of the same thing there would be more than one definition and more than one nature, since, starting from different parts, a same number would come about in different ways. Deinde cum dicit: in plus autem semper est intelligere etc., assignat causam quare in numeris additio excedit omnem determinatam multitudinem. Et dicit quod possumus semper intelligere quolibet numero dato alium maiorem, per hoc quod magnitudo dividitur in infinitum. Manifestum est enim quod divisio causat multitudinem: unde quanto plus dividitur magnitudo, tanto maior multitudo consurgit; et ideo ad infinitam divisionem magnitudinum sequitur infinita additio numerorum. Et ideo sicut infinita divisio magnitudinis non est in actu sed in potentia, et excedit omne determinatum in minus, ut dictum est; ita additio numerorum infinita non est in actu sed in potentia, et excedit omnem determinatam multitudinem. Sed hic numerus, qui sic in infinitum multiplicatur, non est numerus separatus a decisione magnitudinum. 393. Then  he gives the reason why in numbers addition exceeds any predetermined multitude. And he says that we can always think of a number greater than any given number, for the reason that magnitude is divided to infinity. For it is plain that division causes multitude; hence the more magnitude is divided the greater is the multitude that results, and upon the infinite division of magnitudes there follows the infinite addition of numbers. Therefore just as infinite division of magnitude is not in act but in potency, and exceeds every determinate quantity in smallness, as was said (nos. 391,392), so the infinite addition of numbers is not in act but in potency, and exceeds every determinate multitude. But this number which is thus multiplied to infinity is not a number independent of the division of magnitudes. Circa quod sciendum est quod divisio, ut dictum est, multitudinem causat. Est autem duplex divisio: una formalis, quae est per opposita; et alia secundum quantitatem. Prima autem divisio causat multitudinem, quae est de transcendentibus, secundum quod ens dividitur per unum et multa: sed divisio continuae quantitatis causat numerum, qui est species quantitatis, inquantum habet rationem mensurae. Et hic numerus multiplicabilis est in infinitum, sicut et magnitudo divisibilis est in infinitum: sed multitudo quae sequitur divisionem formalem rerum, non multiplicatur in infinitum; sunt enim determinatae species rerum, sicut et determinata quantitas universi. Et ideo dicit quod hic numerus, qui multiplicatur in infinitum, non separatur a divisione continui. Neque tamen hic numerus sic est infinitus, sicut aliquid permanens, sed sicut semper in fieri existens, inquantum successive additur supra quemlibet numerum datum; sicuti etiam est de tempore et de numero temporis. Numerus enim temporis crescit successive per additionem diei ad diem, non quod omnes dies sint simul. 394. On this point it must be remembered that division, as was stated (no. 393), causes multitude. But division is of two kinds: one is formal, which is through opposites; the other is according to quantity. Now the first division causes that multitude which is a transcendental, accordingly as being is divided into “one” and “many”; but the division of continuous quantity causes number, which is a species of quantity, insofar as it has the notion of measure. And this number can grow to infinity, just as magnitude is divisible to infinity. But the multitude which arises from formal division cannot grow to infinity. For the species of things are determined, just as there is a determined quantity of the universe. That is why he says that the number which grows to infinity is not separated from the division of the continuum. Nor is this number infinite in the sense of something permanent. Rather it is as something always in a state of becoming, inasmuch as, to any given number, additions may be successively made, as is evident in the case of time and the number of time. For the number of time increases successively by the addition of day to day but not all days existed at once. Deinde cum dicit: in magnitudinibus autem etc., ostendit quod e contrario est in magnitudinibus. Dividitur enim continuum in infinitum, ut dictum est. Sed in maius non procedit in infinitum etiam secundum potentiam, quia quantum unumquodque est in potentia, tantum potest esse in actu. Si igitur esset in potentia naturae quod cresceret aliqua magnitudo in infinitum, sequeretur quod esset aliqua magnitudo sensibilis infinita; quod est falsum, ut supra dictum est. Relinquitur igitur quod non est in potentia additio magnitudinum in infinitum sic quod excedatur omnis determinata quantitas: quia sequeretur quod esset aliquid maius caelo. 395. Then  he shows that the opposite occurs in magnitudes. For although a continuum be divided to infinity, as was said (nos. 393,394), the size cannot grow indefinitely even potentially. For as great as a thing is in potency, so great can it be in act. If, therefore, it were in the potency of nature that a magnitude grow to infinity, it would follow that there would actually be some infinite sensible magnitude—which is false, as stated (ll.8.9). The consequence is, therefore, that addition of magnitudes cannot go on to infinity so as to exceed every pre-determined quantity; for otherwise there would be something greater than the heavens. Ex quo patet falsum esse, quod quidam dicunt, quod in materia prima est potentia ad omnem quantitatem: non enim est in materia prima potentia nisi ad determinatam quantitatem. Patet etiam ex praemissis ratio quare non oportet numerum tantum esse in actu, quantum est in potentia, sicuti hic dicitur de magnitudine: quia additio numeri sequitur divisionem continui, per quam a toto itur ad id quod est in potentia ad numerum. Unde non oportet devenire ad aliquem actum finientem potentiam. Sed additio magnitudinis ducit in actum, ut dictum est. 396. From the foregoing it is plain that the claim of some that in prime matter there is a potency to every quantity is false; for in prime matter there is a potency only to determined quantity. It Is plain also from the foregoing why number does not have to be as great in act as it is potentially, as is said here of magnitude: for addition occurs in number as a consequence of the division of the continuum, by which one passes from a whole to what is in potency to number. Hence one need not arrive at some act terminating the potency. But the addition of magnitudes arrives at act, as was said (no. 391). Commentator autem assignat aliam rationem: quia potentia ad additionem magnitudinis est in una et eadem magnitudine; sed potentia ad additionem numerorum est in diversis numeris, inquantum cuilibet numero potest aliquid addi. Sed haec ratio parum valet, quia sicut per additionem est alia et alia species numeri, ita alia et alia species mensurae, secundum quod bicubitum et tricubitum dicuntur species quantitatis. Et etiam quidquid additur superiori numero, additur inferiori; et secundum hoc in uno et eodem numero, scilicet binario vel ternario, est potentia ad infinitam additionem. The Commentator [Averroes], however, assigns another reason: namely, that potency to addition in magnitude is in one and the same magnitude but the potency to addition in numbers is in various numbers inasmuch as to any number something can be added. But this reason has little value because just as addition produces varying species of number, so also varying species of measure, as, for example, “two cubits long” and “three cubits long” are called species of quantity. Moreover, whatever is added to a higher number is added to the lower. Accordingly, there is in one and the same number, e.g., two or three, a potency to infinite addition. Deinde cum dicit: infinitum autem non idem est etc., ostendit quomodo infinitum inveniatur diversimode in diversis. Et dicit quod infinitum non est secundum eandem rationem in motu et in magnitudine et tempore, ac si esset una natura univoce praedicata de eis: sed dicitur de posteriori eorum secundum prius, sicut de motu propter magnitudinem, in qua est motus, vel localis vel alterationis vel augmenti; de tempore autem propter motum. Et hoc ideo quia infinitum competit quantitati, motus autem est quantus secundum magnitudinem, et tempus propter motum, ut infra patebit. Et ideo dicit quod nunc utimur his, sed posterius manifestabitur de unoquoque eorum quid sit, et quod omnis magnitudo sit divisibilis in magnitudines. 397. Then  he shows how the infinite is found in diverse ways in diverse things. And he says that the infinite is not found according to the same aspect in motion and magnitude and time, as if it were one nature being predicated univocally in all three cases. Rather it is said of the subsequent member in terms of its antecedent, for example, of motion by reference to the magnitude in which notion takes place (whether it be local motion, alteration or augmentation;) and of time by reference to notion. This happens because the infinite pertains to quantity, and notion is quantified by reference to magnitude, while time is quantified by reference to motion, as will be evident below (Bk. IV, l.17). And therefore he says that we are now mentioning these, but later what each of them is will be explained, as well as that every magnitude is divisible into magnitudes (Bk. VI, l.1). Deinde cum dicit: non removet autem ratio mathematicos etc., ostendit quomodo mathematici utuntur infinito. Et dicit quod ratio praedicta, qua ponimus non esse magnitudinem infinitam in actu, non removet considerationem mathematicorum, qui utuntur infinito; puta cum geometra dicit, sit talis linea infinita. Non enim indigent ad suam demonstrationem infinito in actu, neque eo utuntur: sed solum indigent quod sit aliqua linea finita tanta quanta est eis necessaria, ut ex ea possint subtrahere quod volunt. Et ad hoc sufficit quod aliqua maxima magnitudo sit; quia alicui maximae magnitudini competit, quod possit dividi secundum quantamcumque proportionem respectu alterius magnitudinis datae. Unde ad demonstrandum non differt utrum sit hoc modo vel illo, scilicet vel infinita vel finita maxima quantitas. Sed quantum ad esse rei multum differt, utrum sit vel non sit. 398. Then  he explains how mathematicians make use of the infinite, and says that the argument that there is no actually infinite magnitude,(ll.8,9) does not destroy the consideration of the mathematicians, who use the infinite, as, for example, when the geometer says, “Let this line be infinite.” For they do not need for their demonstrations the infinite in act, nor do they use it, but they need only some finite line of sufficient quantity for their needs, so as to be able to subtract from it so much as they wish. For their purpose it is enough that there exist some maximum magnitude which can be divided according to any proportion in respect to another given magnitude. Hence, for purposes of demonstration, it makes no difference whether this maximum magnitude be one way or the other, i.e., finite or infinite; but as to the being of things, it makes a great difference whether it is one or the other. Deinde cum dicit: quoniam autem causae divisae sunt etc., ostendit quomodo infinitum sit principium. Et dicit quod cum sint quatuor genera causarum, ut supra dictum est, patet ex praemissis quod infinitum est causa sicut materia: infinitum enim habet esse in potentia, quod est proprium materiae. Sed materia quidem quandoque est sub forma, quandoque autem sub privatione. Infinito autem non competit ratio materiae secundum quod est sub forma, sed secundum quod est sub privatione: quia scilicet infinitum dicitur per remotionem perfectionis et termini. Et propter hoc subiungit quod ipsi infinito esse est privatio, idest ratio infiniti in privatione consistit. 399. Then  he shows how the infinite is a principle. And he says that since there are four genera of causes, as was said above (Bk. II), the infinite is a cause in the manner of matter. For the infinite has being in potency, which is proper to matter. Now matter is sometimes under a form and sometimes under privation. The infinite, however, has the notion of matter, not insofar as matter lies under a form but inasmuch as matter has privation—for the infinite implies the lack of perfection and term. That is why the Philosopher adds that the being of the infinite is privation, i.e., the notion of the infinite consists in privation. Et ne aliquis intelligat quod infinitum est materia sicut materia prima, subiungit quod per se subiectum privationis, quae constituit rationem infiniti, est continuum sensibile. Et hoc apparet, quia infinitum quod est in numeris causatur ex infinita divisione magnitudinis; et similiter infinitum in tempore et motu causatur ex magnitudine: unde relinquitur quod primum subiectum infiniti sit continuum. Et quia magnitudo secundum esse non est separata a sensibilibus, sequitur quod subiectum infiniti sit sensibile. And lest anyone suppose that the infinite is matter like prime matter, he adds that the per se subject of the privation which constitutes the nature of the infinite is the sensible continuum. That this is so is clear from the fact that the infinite found in numbers is caused from the infinite division of magnitude; and similarly, the infinite in time and notion are caused by magnitude. Hence, the first subject of the infinite in the continuum. And since really existing magnitude is not separated from sensible things, it follows that the subject of the infinite is sensible. Et in hoc etiam concordant omnes antiqui, qui utuntur infinito sicut principio materiali. Unde inconveniens fuit quod attribuerunt infinito continere, cum materiae non sit continere, sed magis contineri. And on this point all the earlier philosophers agree who use the infinite as a material principle. Wherefore they improperly attributed to the infinite the capacity to contain, for matter does not contain but rather is contained.
Solution of arguments in favor of existence of the infinite
Chapter 8 Λοιπὸν δ' ἐπελθεῖν καθ' οὓς λόγους τὸ ἄπειρον εἶναι δοκεῖ οὐ μόνον δυνάμει ἀλλ' ὡς ἀφωρισμένον· τὰ μὲν γάρ ἐστιν αὐτῶν οὐκ ἀναγκαῖα, τὰ δ' ἔχει τινὰς ἑτέρας ἀληθεῖς ἀπαντήσεις. It remains to dispose of the arguments which are supposed to support the view that the infinite exists not only potentially but as a separate thing. Some have no cogency; others can be met by fresh objections that are valid. οὔτε γὰρ ἵνα ἡ γένεσις μὴ ἐπιλείπῃ, ἀναγκαῖον ἐνεργείᾳ ἄπειρον εἶναι σῶμα αἰσθητόν· ἐνδέχεται γὰρ τὴν θατέρου φθορὰν θατέρου εἶναι γένεσιν, πεπερασμένου ὄντος τοῦ παντός. (1) In order that coming to be should not fail, it is not necessary that there should be a sensible body which is actually infinite. The passing away of one thing may be the coming to be of another, the All being limited. ἔτι τὸ ἅπτεσθαι καὶ τὸ πεπεράνθαι ἕτερον. τὸ μὲν γὰρ πρός τι καὶ τινός (ἅπτεται γὰρ πᾶν τινός), καὶ τῶν πεπερασμένων τινὶ συμβέβηκεν, τὸ δὲ πεπερασμένον οὐ πρός τι· οὐδ' ἅψασθαι τῷ τυχόντι τοῦ τυχόντος ἔστιν. (2) There is a difference between touching and being limited. The former is relative to something and is the touching of something (for everything that touches touches something), and further is an attribute of some one of the things which are limited. On the other hand, what is limited is not limited in relation to anything. Again, contact is not necessarily possible between any two things taken at random. τὸ δὲ τῇ νοήσει πιστεύειν ἄτοπον· οὐ γὰρ ἐπὶ τοῦ πράγματος ἡ ὑπεροχὴ καὶ ἡ ἔλλειψις, ἀλλ' ἐπὶ τῆς νοήσεως. ἕκαστον γὰρ ἡμῶν νοήσειεν ἄν τις πολλαπλάσιον ἑαυτοῦ αὔξων εἰς ἄπειρον· ἀλλ' οὐ διὰ τοῦτο ἔξω [τοῦ ἄστεός] τίς ἐστιν [ἢ] τοῦ τηλικούτου μεγέθους ὃ ἔχομεν, ὅτι νοεῖ τις, ἀλλ' ὅτι ἔστι· τοῦτο δὲ συμβέβηκεν. (3) To rely on mere thinking is absurd, for then the excess or defect is not in the thing but in the thought. One might think that one of us is bigger than he is and magnify him ad infinitum. But it does not follow that he is bigger than the size we are, just because some one thinks he is, but only because he is the size he is. The thought is an accident. ὁ δὲ χρόνος καὶ ἡ κίνησις ἄπειρά ἐστι καὶ ἡ νόησις οὐχ ὑπομένοντος τοῦ λαμβανομένου. (a) Time indeed and movement are infinite, and also thinking, in the sense that each part that is taken passes in succession out of existence. μέγεθος δὲ οὔτε τῇ καθαιρέσει οὔτε τῇ νοητικῇ αὐξήσει ἔστιν ἄπειρον. ἀλλὰ περὶ μὲν τοῦ ἀπείρου, πῶς ἔστι καὶ πῶς οὐκ ἔστι καὶ τί ἐστιν, εἴρηται. (b) Magnitude is not infinite either in the way of reduction or of magnification in thought. This concludes my account of the way in which the infinite exists, and of the way in which it does not exist, and of what it is.
Postquam philosophus per definitionem infiniti assignavit rationes eorum quae de infinito dicuntur, hic solvit rationes quae supra positae sunt ad ostendendum infinitum esse. 400. After the Philosopher has used the definition of the infinite to explain the things attributed to it he now solves the argument presented above (l.7) to show the infinite existed. Et primo dicit de quo est intentio; secundo exequitur propositum, ibi: neque enim ut generatio et cetera. First, he proposes his intention; Then he follows it out, at 401. Dicit ergo primo quod post ea quae dicta sunt de infinito, reliquum est solvere rationes secundum quas videbatur ostendi quod infinitum sit non solum in potentia, sicut supra determinavimus, sed quod sit in actu, sicut ea quae sunt finita et determinata. Aliquae enim illarum rationum non concludunt ex necessitate, sed sunt totaliter falsae; aliquae autem earum ex aliqua parte verum concludunt. He says therefore first  that after speaking of the nature of the it remains to settle the arguments which appeared to show that the infinite is not only something in potency, as we determined above (l. 10), but that was in act, as things are that are finite and determined. For some of the arguments do not conclude necessarily but are entirely false, while others are partially true. Deinde cum dicit: neque enim ut generatio etc., solvit quinque rationes quae supra positae sunt ad ostendendum infinitum esse. Et primo solvit eam quae sumebatur ex parte generationis. Concludebatur enim quod si generatio non deficit, quod oporteat esse infinitum. Sed haec ratio quantum ad hoc verum concludit, quod infinitum sit in potentia, quae successive in actum reducatur, sicut supra dictum est. Sed non est necessarium quod sit aliquod corpus sensibile infinitum in actu, ad hoc quod generatio non deficiat, sicut antiqui aestimaverunt, ponentes in infinitum conservari generationem, ac si semper generatio fieret per extractionem ex aliquo corpore; quod in infinitum fieri non posset nisi illud corpus esset infinitum. Sed hoc non est necessarium; cum toto corpore sensibili existente finito, generatio in infinitum durare possit per hoc quod corruptio unius est generatio alterius. 401. Then  he solves the five reasons cited above (l.7) as proving that the infinite exists. And first he solves the one based on the fact of generation. For it concluded that if generation does not cease, then the infinite must be. Now this argument concludes truly insofar as the infinite is in a potency that is successively reduced to act. But it is not necessary that there be some sensible body which is infinite in act, in order to account for generation not ceasing, as the earlier philosophers supposed when they said that generation continues to infinity, supposing it to take place by extracting from some body, with the consequence that the process could not be infinite unless that body were infinite. But this is not necessary: for even supposing the whole of sensible body as finite, generation can endure ad infinitum by the fact that the corruption of one thing is the generation of another. Deinde cum dicit: amplius tangi et includi etc., solvit rationem quae sumebatur ex parte contactus; ac si necessarium sit omne corpus finitum tangere quoddam aliud; et sic oporteat in infinitum procedere. Sed ipse solvit, quod alterum est tangi et finiri: quia tangi et includi dicitur respectu alterius; omne enim tangens tangit aliquid: sed finitum dicitur absolute, et non ad aliud, inquantum per proprios terminos aliquid finitum est in seipso. Accidit enim alicui finito quod tangat. Non tamen oportet quod omne tactum ab uno tangat aliud; ut sic in infinitum procedatur. Unde manifestum est quod haec ratio omnino nihil ex necessitate concludit. 402. Then  he solves the argument based on the principle of contact, as though it were necessary for every finite body to touch some other body and so on to infinity. But he solves this by saying that it is one thing to be “touched” and another to be “terminated”, because to be “touched” and “enclosed” are said in respect to something else, for whatever touches, touches something else. To be “terminated,” however, is said absolutely and does not imply a relationship to something else, because a thing is made finite in itself by its own terminations. For it is incidental to the finite that it be touching something. Nevertheless, neither is it necessary that everything touched by something should touch something else and that this go on to infinity. Hence it is evident that this argument does not conclude anything of necessity. Deinde cum dicit: intelligentiae autem credere etc., solvit rationem quae sumitur ex parte intellectus et imaginationis, quam antiqui non distinguebant ab intellectu. Per hanc autem rationem supra ostendebatur quod esset spatium infinitum extra caelum, et per consequens locus et corpus. Sed ipse dicit quod inconveniens est credere intelligentiae, ita scilicet quod quidquid apprehenditur intellectu vel imaginatione sit verum, ut quidam antiquorum putaverunt, quorum opinio reprobatur in IV Metaphys. Non enim sequitur, si apprehendo aliquam rem minorem vel maiorem quam sit, quod sit aliqua abundantia vel defectus in re illa, sed solum in apprehensione intellectus vel imaginationis. Potest enim aliquis intelligere quemcumque hominem esse multiplicem eius quod est, idest duplum vel triplum vel qualitercumque augmentans in infinitum: non tamen propter hoc erit aliqua huiusmodi quantitas multiplicata extra intellectum, aut extra determinatam quantitatem aut magnitudinem: sed contingit quod re sic existente, aliquis ita intelligat. 403. Then  he solves the argument based on the intellect and the imagination, which latter the ancients did not distinguish from the intellect. This argument above (l.7) concluded that there was outside the universe an infinite space, and consequently a place and a body. But it is incorrect to “trust to thought,” i.e., believe that whatever is apprehended by the imagination or intellect is true, as some of the ancients thought, whose opinion is refuted in Metaphysics IV. For if I apprehend a thing as smaller or larger than it is, it does not thereby follow that there is such an abundance or defect in the object itself but only in the apprehension of the intellect or imagination. For one might understand some man to be a multiple of himself, i.e., two or three times larger than he really if, or any other amount to infinity, yet there will not be because of this a corresponding multiplication of him outside the intellect or outside a definite quantity or magnitude. Deinde cum dicit: tempus autem et motus etc., solvit rationem acceptam ex tempore et motu. Et dicit quod tempus et motus sunt infinita non in actu, quia nihil est temporis in actu nisi nunc; neque aliquid motus est in actu nisi quoddam indivisibile: sed intellectus apprehendit continuitatem temporis et motus, accipiendo ordinem prioris et posterioris: ita tamen quod id quod primo fuit acceptum de tempore vel motu, non permanet sic. Unde non oportet dicere quod totus motus infinitus sit in actu, vel totum tempus infinitum. 404. But while a thing remains what it is, one can conceive of it in such a manner. Then  he solves the difficulty based on time and motion. And he says that time and motion are not infinite in act, because nothing of time is actual but the “now,” and nothing of motion is actual except a kind of indivisible. But the intellect apprehends a continuity in time and in motion by apprehending an order of “prior” and “posterior,” in such a way, however, that what was first taken in time or in motion does not remain in the same state. Hence it is not necessary to say that the whole of motion is infinite, or that the whole of time is infinite. Deinde cum dicit: magnitudo autem neque divisione etc., solvit rationem sumptam ex parte magnitudinis. Et dicit quod magnitudo non est infinita in actu neque per divisionem neque per augmentationem intelligibilem, sicut ex supra dictis patet. 405. Then  he solves the argument based on magnitude, and he says that magnitude is not infinite in act either as a result of division or of an intelligible increase, as is evident from what was said above (ll.8-10). Ultimo autem epilogat quod dictum est de infinito. Finally he summarizes by saying that we have completed our study of the infinite.