ARISTOTLE 206a καὶ ἐνεργείᾳ: ᾿Ολύμπια yap ἔστι καὶ τῷ δύνασθαι a τὸν ἀγῶνα γίνεσθαι καὶ τῷ γίνεσθαι. ἤάλλως δ᾽ ἔν τε τῷ χρόνῳ δῆλον τὸ ἄπειρον καὶ Ὁ χρόνῳ δῆ ρ ἐπὶ τῶν ἀνθρώπων καὶ ἐπὶ τῆς διαιρέσεως τῶν ἐεγεθῶν. ὅλως μὲν γὰρ οὕτως ἐστὶ τὸ ἄπειρον, τῷ ἀεὶ ἄλλο καὶ ἄλλο λαμβάνεσθαι, καὶ τὸ λαμ- βανόμενον μὲν ἀεὶ εἶναι πεπερασμένον, ἀλλ᾽ ἀεί γε ἕτερον καὶ ἕτερον' [ἔτι τὸ εἶναι πλεοναχῶς 80 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, εἰ καὶ πεπερασμένον, ἀλλ᾽ ἀεί ye ἕτερον καὶ ἕτερον] ἀλλ᾽ 806} ἐν μὲν τοῖς μεγέθεσιν ὑπομένοντος τοῦ ληφθέντος τοῦτο συμβαίνει, ἐπὶ δὲ τοῦ χρόνου καὶ τῶν ἀνθρώπων φθειρομένων οὕτως ὥστε μὴ ἐπιλείπειν. ‘To δὲ κατὰ πρόσθεσιν τὸ αὐτό ἐστί πὼς καὶ τὸ κατὰ διαίρεσιν" ἐν γὰρ τῷ πεπερασμένῳ κατὰ ὃ πρόσθεσιν “γίνεται ἀντεστραμμένως" ἡ γὰρ διαιρού- μενον ὁρᾶταν εἰς ἄπειρον, ταύτῃ προστιθέμενον φανεῖται πρὸς τὸ ὡρισμένον. ἂν γὰρ τῷ πεπερα- σμένῳ μεγέθει. ἂν λαβών τις ὡρισμένον ᾿προσλαμ- βάνῃ τῷ αὐτῷ λόγῳ, μὴ τὸ αὐτό τι τοῦ ὅλου μέγεθος περιλαμβάνων, οὐ διέξεισι τὸ πεπερα- 10 σμένον" ἐὰν δ᾽ οὕτως αὔξῃ τὸν λόγον ὥστε ἀεί τι τὸ αὐτὸ περιλαμβάνειν μέγεθος, διέξεισι, διὰ τὸ 1 [ἔτι τὸ εἶναι πλεοναχῶς λέγεται Simpl. 495. 6, Philop. 468, 8 (lemma): ὅτι τὸ εἶναι πλεοναχῶς λέγεται EL: om. cett. Philop. excised this sentence as not contained ‘in the most accurate copies,’ Diels (Zur Textgesch. p. 32), held it to be a duplicate of ἀλλ ἐπεὶ πολλαχῶς.,, ἕτερον Oi 21-29), from another recension. Cf, Simplic, loc, cit.— PHYSICS, ITT. νι, events in question is not (like the statue-potentialities of the bronze) all actualized at once, but is in course of transit into and out of actuality as long as it lasts. The Olympic games, as a whole, are a potentiality only, even when they are in process of actualization, Moreover, ‘ having no limit’ is not quite the same thing as applied to time or to the human race ¢ and as applied to the possibility of continuously dividing a ‘magnitude,’ as it decreases. In all these cases, the ‘ absence of limit’ may be regarded as the open ‘ possibility of more,’ the ‘more’ that is actually taken being always limited, but always different; but when this occurs in the case of magnitudes what is once taken remains; whereas in the case of time or of the human race the parts taken are constantly perishing in such a way that the succession never fails.® There is a certain process of unlimited addition that can be identified by reciprocity with un- limited division; for as we see the finite magnitude in process of division ad infinitum, so we shall find the process of addition tending to a definite limit. For if (1) one should take a definite piece away from a limited magnitude and then go on to take away the same proportion of what is left (not the same fraction of the original whole), and so on and so on, he will never work through to the end of the original magnitude; whereas, if (2) he increases the proportion of the remainder which he takes away each time, so as to make the actual magnitude taken away always the same, then he will get 4 [The human race may never perish, but the individuals (or generations) of which it consists are always perishing.—C. ] > [μὴ ἐπιλείπειν, Cf. 208 a 8 ἵνα ἡ γένεσις μὴ ἐπιλείπῃ.---(,] ARISTOTLE 206b πᾶν τὸ πεπερασμένον ἀναιρεῖσθαι ὁτῳοῦν ὧρι- σμένῳ, ἄλλως μὲν οὖν οὐκ ἔστιν, οὕτως δ᾽ ἔστι τὸ ἄπειρον, δυνάμει τε καὶ ἐπὶ καθαιρέσει. καὶ ἐντελεχείᾳ δέ ἐστιν ὡς τὴν ἡμέραν εἶναι λέγομεν kat τὸν ἀγῶνα, καὶ δυνάμει οὕτως ὡς ἡ ὕλη, καὶ οὐ καθ᾽ αὑτὸ ὡς τὸ πεπερασμένον. καὶ κατὰ πρόσθεσιν δὴ οὕτως ἄπειρον δυνάμει ἐστίν, ὃ ταὐτὸ λέγομεν τρόπον τινὰ εἶναι τῷ κατὰ διαίρεσιν’ ἀεὶ μὲν γάρ τι αὐτοῦ ἔξω ἔσται λαμβάνειν, οὐ μέντον ὑπερβαλεῖ παντὸς ὡρισμένου μεγέθους, ὧσ- ἢ περ ἐπὶ τὴν διαίρεσιν ὑπερβάλλει παντὸς wpt- σμένου καὶ ἀεὶ ἔσται ἔλαττον. 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 28 TL τοιοῦτον, ἄπειρον εἶναι. ἀλλ᾽ εἰ μὴ οἷόν τε εἶναι ἄπειρον ἐντελεχείᾳ σῶμα αἰσθητὸν οὕτω, φανερὸν ὅτι οὐδὲ δυνάμει ἂν εἴη κατὰ πρόσθεσιν, ἀλλ᾽ ἣ * For instance, let AO be a finite line divided at B, C, D, cte., so that Ἢ τ πον ἘΞ ΕΝ cD Ve ait so on and so on.
A B Cc Dd fe) Then the subtraction of AB, BC,...ete, frum AO can go on for ever and there will always be some of AO left. And the addition of AB, BC, ete,, can go on for ever withoul completing AO. But if the points are so placed that AB = BC =CD A B [οὶ D then the subtraction of AB, BC, and CD will exhaust AO and come to an end. And the addition of AB, BC, and CD will come to an end by completing AQ, PHYSICS, III. vr.
through to the end; for successive withdrawals of any constant magnitude, however small, will exhaust any limited magnitude whatever. The Unlimited, then, exists only the way just described—as an unlimited potentiality of approximation by reduction of intervals. Unlimitedness is never actual except in the sense in which we can say ‘ the day’ or ‘ the games’ are actual, whereas as potentiality it is analogous to formless matter; it never exists as a thing, as a determined quantum does. Accordingly, in this sense—as a potentiality—there is also an infinite by way of addition, the thing we described as being in a way the same as the infinite by way of division, for in addition it will always be possible to find something beyond the total for the time being (in a convergent series) though the total will never exceed every assigned magnitude in the way that, in the direction of division, the result does pass every assigned magnitude and will always become still smaller. But, in the sense of exceeding every finite magnitude as the result of addition, the unlimited cannot exist even potentially, if we reject the hypothesis of the physicists who suppose some such actual substance as air or the like to have bodily existence outside the universe and to be unlimited, In that case indeed ‘ infinity’ would incidentally have an actual existence, though not itself substantial. But if (as has been shown) it is impossible that there should exist any such sensible Body, as an accomplished actuality, it follows thal there is not so much as a potentiality of a sum of additions extending beyond all assignable magnitude. The only potentiality of unlimited additions, then, is the additions that are the ARISTOTLE 206} ὥσπερ εἴρηται ἀντεστραμμένως τῇ διαιρέσει' ἐπεὶ καὶ Πλάτων διὰ τοῦτο δύο τὰ ἄπειρα ἐποίησεν, ὅτι καὶ ἐπὶ τὴν αὔξην δοκεῖ ὑπερβάλλειν καὶ εἰς ? 80 ἄπειρον ἰέναι καὶ ἐπὶ τὴν καθαίρεσιν. ποιήσας μέντοι δύο οὐ χρῆται: οὔτε γὰρ ἐν τοῖς ἀριθμοῖς ᾽ ε A τὸ ἐπὶ τὴν καθαίρεσιν ἄπειρον ὑπάρχει (ἡ yap μονὰς ἐλάχιστον) οὔτε ἐπὶ τὴν αὔξην (μέχρι yap δεκάδος ποιεῖ τὸν ἀριθμόν). Συμβαίνει δὲ τοὐναντίον ἄπειρον εἷναι ἢ ὡς 9014 λέγουσιν: οὐ γὰρ οὗ μηδὲν ἔξω, ἀλλ᾽ οὗ ἀεί τι ἔξω ἔστι, τοῦτο ἄπειρόν ἐστιν. σημεῖον δέ: καὶ γὰρ τοὺς δακτυλίους ἀπείρους λέγουσι τοὺς μὴ Ν / uv > ἢ μὲ μ / ἔχοντας σφενδόνην, ὅτι ἀεί τι ἔξω ἔστι λαμβάνειν, καθ᾽ ὁμοιότητα μέν τινα λέγοντες, οὐ μέντοι 5 κυρίως" δεῖ γὰρ τοῦτό τε ὑπάρχειν καὶ μηδέποτε τὸ αὐτὸ λαμβάνεσθαι, ἐν δὲ τῷ κύκλῳ οὐ γίνεται οὕτως GAN’ del τὸ ἐφεξῆς μόνον ἕτερον. ἄπειρον μὲν οὖν ἐστιν οὗ κατὰ ποσὸν λαμβάνουσιν ἀεί τι λαβεῖν ἔστιν ἔξω' οὗ δὲ μηδὲν ἔξω, τοῦτ᾽ ἐστὶ “λ \ δλ A ¢ ζό θ + ὅλ τέλειον καὶ ὅλον. οὕτω yap δριζόμεθα τὸ ὅλον, 10 οὗ μηθὲν ἄπεστιν, οἷον ἄνθρωπον ὅλον ἢ κιβωτόν" A ὥσπερ δὲ τὸ καθ᾽ ἕκαστον, οὕτω καὶ τὸ κυρίως, > (The Platonists adopted from the Pythagoreans the doctrine that all numbers are reducible to the numbers from one to ten, because after ten we go back to 1, 2, 3, etc. Theon Smyrn. p. 162 (Dupuis), Met. 1073 a 20, 1084 a 12.
3 bn circular movement see Book VIII. chap. ix.
PHYSICS, III. νι.
obverse of the subtractions regulated by successive divisions as already explained. And this is why Plato himself distinguishes between two ‘infinities,’ thinking that he must have an infinite that could exceed all expansion as well as all reduction. But although he postulates such an infinity, he never makes any use of it; for in numbers he does not admit either reduction without limit (since the monad is the irreducible minimum), or increase without limit, for the series of numbers stops at the decad.? The fact is that the unlimited is really the exact opposite of its usual description; for 1t is not that ‘beyond which there is nothing,’ but that ‘ of which there is always more beyond.’ And this is really implied in rings that have no gem-sockets being called ‘ endless,’ because wherever you are you go on to more. But the analogy is not complete, for if anything is really to be ‘ endless * you must be able to go on, not only to ‘ more,’ but to what you had never gone over or done before; and this is not so with the ring, for you keep on covering the same part of it once more and once more, and it is only the next point that is always different from the one before it.¢ The unlimited, then, is that of which, however much you have taken, there is always more to take; that of which there is nothing more to take is not unlimited, but whole or completed. For we define a whole precisely as that from which nothing is absent, for example, a ‘whole man’ or a ‘ whole chest.’ Aid as with particular wholes, so when the word is used in the strict sense: ὦ the Whole is that 4 fAristotle refers to the use of τὸ ὅλον and τὸ πᾶν to mean ‘the universe ’—ihe whole ‘in the strict sense,’ be- cause il is not also a part of any larger whole.—C.]
16 20 2h 80 ARISTOTLE οἷον τὸ ὅλον od μηδέν ἐστιν ἔξω: od δ᾽ ἐστὶν ἀπουσία ἔξω, οὐ πᾶν, 6 τι ἂν ἀπῇ. ὅλον δὲ καὶ τέλειον ἢ τὸ αὐτὸ πάμπαν ἢ σύνεγγυς τὴν φύσιν ἐστίν, τέλειον δ᾽ οὐδὲν μὴ ἔχον τέλος’ τὸ δὲ τέλος πέρας.
Διὸ βέλτιον οἰητέον Παρμενίδην Μελίσσου εἰρη- 4 Ly 4 A av \ μὲ a c 4 κέναι' ὃ μὲν yap ἄπειρον τὸ ὅλον φησίν, ὁ δὲ τὸ ὅλον πεπεράνθαι ᾿ μεσσοθεν ἰσοπαλές. od yap λίνον λίνῳ συνάπτειν ἐστὶ τῷ ἅπαντι καὶ ὅλῳ τὸ ἄπειρον' ἐπεὶ ἐντεῦθέν ye λαμβάνουσι τὴν σεμνόa, A τῆτα κατὰ τοῦ ἀπείρου---τὸ ‘ πάντα περιέχειν ’ a ε - καὶ τὸ “πᾶν ἐν ἑαυτῷ ἔχειν ᾿--διὰ τὸ ἔχειν τινὰ ὁμοιότητα τῷ ὅλῳ. ἔστι γὰρ τὸ ἄπειρον τῆς τοῦ μεγέθους τελειότητος ὕλη Kal τὸ δυνάμει ὅλον, ἐντελεχείᾳ δ᾽ οὔ, διαιρετὸν δ᾽ ἐπί τε τὴν καθαίρεσιν καὶ τὴν ἀντεστραμμένην πρόσθεσιν: ὅλον δὲ καὶ πεπερασμένον οὐ Kal’ αὐτὸ ἀλλὰ κατ᾽ ἄλλο. καὶ οὐ περιέχει ἀλλὰ περιέχεται, ἦ ἄπειρον. διὸ καὶ ἄγνωστον 4 ἄπειρον' εἶδος yap οὐκ ἔχει ἡ ὕλη. wore φανερὸν ὅτι μᾶλλον ἐν μορίου λόγῳ τὸ ἄπειρον ἢ ἐν ὅλου: μόριον yap ἡ ὕλη τοῦ ὅλου ὥσπερ ὁ χαλκὸς τοῦ χαλκοῦ ἀνδριάντος. ἐπεὶ εἴ A a 3 a a ye περιέχει ἐν τοῖς αἰσθητοῖς, καὶ ἐν τοῖς νοητοῖς τὸ μέγα καὶ τὸ μικρὸν ἔδει περιέχειν τὰ νοητά' 4 [Parm. 8. 42 αὐτὰρ ἐπεὶ πεῖρας πύματον τετελεσμένον ἐστί πάντοθεν εὐκύκλου σφαίρης ἐναλίγκιον ὄγκῳ, | μεσσόθεν ἰσοπαλὲς πάντῃ.--Ο,Ἶ] PITYSICS, IIT. νι.
outside which there is nothing whatsoever; whereas that from which something, no matter what, is miss- ing and left outside is not ‘ All.’ And ‘ whole’ and ‘complete,’ if not absolutely the same, are very closely akin, and nothing is complete (éeleios) unless it has an end (telos); but an end is a limit.
So Parmenides was nearer the mark than Melissos; for Melissos speaks of ‘ the Whole’ as ‘ unlimited,’ whereas Parmenides sets boundaries to his ‘ whole,’ that is ‘ equipoised on the centre.’* For ‘whole’ or ‘ all’ and ‘ unlimited’ are terms that cannot run in double harness. What led them to give to the Unlimited the impressive attributes of ‘ all-em- bracing ’ and ‘ all in itself containing’ was the fact that it has a certain resemblance to the whole; for the ‘unlimited’ is really the ‘material’ out of which a magnitude is completed, and is the potential, though not the realized, whole. It is ‘ divisibility without limit’ in the direction of reduction or con- verse expansion, and is not any determined whole in itself, but only as the unlimited and ‘ material’ factor of the whole which is constituted as such by the limiting and ‘formal’ factor. As ‘unlimited,’ then, it is embraced and not embracing. Therefore qua unlimited it is unknowable, since ‘ material,’ as such, is formless, So the unlimited were evidently better defined as a part than as the whole, in the sense in which ‘ part’ means the ‘ material constituent,’ as bronze is a part or constituent of the bronze statue. For, if in things of sense the undetermined ‘ great or small’ were the continent and not the contained, analogy would demand that in the noetic world also the unintelligible should embrace the Ideas which are the norm of intelligibility; but it is contradictory ARISTOTLE ARISTOTLE » 4 x + a AY ΕΝ 7 iY 2072 ἄτοπον δὲ Kal ἀδύνατον TO ayvwaTav καὶ τὸ ἀόριστον περιέχειν καὶ ὁρίζειν.
CHAPTER VII ARGUMENT [Why number has a lower limit (‘one’), but no upper limit; whereas magnitude ts potentially (though not actually) infinitely divisible, but cannot actually exceed every finite Infinity of time iv derived from infinity of motion, and oon the continuity (infinite divisibility) of magnitude Κατὰ λόγον δὲ συμβαίνει καὶ τὸ κατὰ πρόσθεσιν μὲν μὴ εἶναι δοκεῖν ἄπειρον οὕτως ὥστε παντὸς 86 ὑπερβάλλειν μεγέθους, ἐπὶ τὴν διαίρεσιν δὲ εἶναι" 8010 περιέχεται γὰρ ὡς ἡ ὕλη ἐντὸς καὶ τὸ ἄπειρον, περιέχει δὲ τὸ εἶδος.
Εὐλόγως δὲ καὶ τὸ ἐν μὲν τῷ ἀριθμῷ εἶναι ἐπὶ τὸ ἐλάχιστον πέρας ἐπὶ δὲ τὸ πλεῖον ἀεὶ παντὸς ὑπερβάλλειν πλήθους, ἐπὶ δὲ τῶν μεγεθῶν τούναν- τίον ἐπὶ μὲν τὸ ἔλαττον παντὸς ὑπερβάλλειν ὃ μεγέθους ἐπὶ δὲ τὸ μεῖζον μὴ εἷναι μέγεθος ἄπειρον. αἴτιον δ᾽ ὅτι τὸ ἕν ἐστιν ἀδιαίρετον, 6 τι περ ἂν ἕν ἢ, οἷον ἄνθρωπος εἷς ἄνθρωπος καὶ οὐ πολλοί: 6 δ΄ ἀριθμός ἐστιν ἕνα πλείω καὶ πόσ᾽ ἅττα' ὥστ᾽ ἀνάγκη στῆναι ἐπὶ τὸ ἀδιαίρετον (τὰ γὰρ δύο καὶ τρία παρώνυμα ὀνόματά ἐστιν, PHYSICS, TIT. ντ.--ντι.
and impossible that the unknowable and undefined should embrace and define anything.
CHAPTER VII ARGUMENT (continued) Mathematicians do not require an actual infinite mag- nitude (Ὁ 27-84).
The infinite is ‘ material cause’; its essence, shortage } its subject, the sensible-continuous. Hence tt is that which is embraced or contained rather than the continent Ir follows also from the rationale of infinity that it cannot exceed all magnitude, but depends on the principle of division; for since it is analogous to the ‘material’ it is contained, whereas it is the ‘form’ that is the continent.
It’ is also quite as it should be that in number there should be an inferior limit, whereas it is always possible to transcend any given number, but that in magnitudes, on the other hand, it should always be possible to make the small smaller, but there can be no magnitude of unlimited greatness. The reason is that unity, as unity, is atomic, the human unit, for instance, being one man and not more than one;* whereas number is of more units than one, and specifically of ‘so many,’ so that you cannot go further back than the mdivisible (for ‘two’ and ‘three,’ that is, two ones and three ones, are both numbers, qua more than one, but different * See Gen. Introd. p. Ixxxv, and p. 22, note a. ~ 15 ARISTOTLE ὁμοίως δὲ καὶ τῶν ἄλλων ἀριθμῶν ἕκαστος). ἐπὶ δὲ τὸ πλεῖον ἀεὶ ἔστι νοῆσαι: ἄπειροι yap αἱ διχοτομίαι τοῦ μεγέθους" ὥστε δυνάμει μὲν ἔστιν, ἐνεργείᾳ δ᾽ οὔ, ἀλλ᾽ ἀεὶ ὑπερβάλλει τὸ λαμβανό- μενον παντὸς ὡρισμένου πλήθους. ἀλλ᾽ οὐ χωρι- στὸς ὁ ἀριθμὸς οὗτος τῆς διχοτομίας, οὐδὲ μένει ἡ ἀπειρία ἀλλὰ γίνεται, ὥσπερ Kal ὁ χρόνος καὶ ὁ ἀριθμὸς τοῦ χρόνου. ἐπὶ δὲ τῶν μεγεθῶν τοὐ- ναντίον ἐστίν" διαιρεῖται μὲν γὰρ εἰς ἄπειρα τὸ συνεχές, ἐπὶ δὲ τὸ μεῖζον οὐκ ἔστιν ἄπειρον. ὅσον yap ἐνδέχεται δυνάμει εἶναι, καὶ ἐνεργείᾳ ev- δέχεται τοσοῦτον εἶναι" ὥστε, ἐπεὶ ἄπειρον οὐδέν ἐστι μέγεθος αἰσθητόν, οὐκ ἐνδέχεται παντὸς ὑπερ- βολὴν εἶναι ὡρισμένου μεγέθους" εἴη yap av τι τοῦ οὐρανοῦ μεῖζον. To δ᾽ ἄπειρον οὐ ταὐτὸν ἐν μεγέθει καὶ κινήσει Kal χρόνῳ, ὡς μία tis φύσις, ἀλλὰ τὸ ὕστερον λέγεται κατὰ τὸ πρότερον, οἷον κίνησις μὲν ὅτι τὸ μέγεθος ἐφ᾽ οὗ κινεῦται ἢ ἀλλοιοῦται ἢ αὐξάα [παρώνυμα are defined as things which derive their name from another name, with a difference of termination, 6.0. ‘grammarian’ from ‘giammar’ (Caf. 1 awll) ‘Two?’ and ‘Three’ (etc.) might be Heed as the dual and plural (grammatical ‘ numbers ’) of ‘ one.’ Simplic. 505. 18 explains that every number derives its name from the number of ones it contains: ‘two’ means ‘ that which consists of two ones,’ and so on.—C.]
® See Gen. Introd. pp. Ixxxix f.
PHYSICS, III. vi.
numbers, gua two or three respectively, and so with the rest).¢ But since you can always make another division of a magnitude into two, however many divisions you have already made to get it, you can always conceive a higher number of divisions than any given number however great; consequently the ‘ possibility of more ’ is always there as a potentiality that cannot be exhaustively realized, but can be carried on through a greater than any assignable number of steps. This inexhaustible ‘number,’ how- ever, is not separable from the dichotomy, and its “unlimitedness ’ is not an accomplished thing like the magnitude itself that is the subject of the dichotomies, but is the accompaniment of the process of dichotomy, always in the inaking and never made;° just like time, and the numerical register of time. So number cannot be reduced below unity, but can be increased indefinitely; but the reverse is truc of magnitude; for a continuous magnitude can be divided beyond any given smallness, but cannot be increased above every given greatness. For any magnitude that can exist potentially can exist actually; so since, as we have seen, nothing sense-perceived can be unlimited, a magnitude in excess of every finite magnitude is an impossibility; it would have to transcend the heavens.
Again ‘ unlimitedness ’ does not stand for the same thing in magnitude, inovement, and time, but differs in accordance with their several natures in a deter- mined order of priority. For continuity, in which the unlimited potentialities inhere, has a stable existence in magnitude, but movement (including modification and growth) is continuous because the magnitude over which iL lakes place is so, and lime ARISTOTLE 207b25 veTal, ὅ χρόνος δὲ διὰ τὴν κίνησιν. νῦν μὲν οὖν χρώμεθα τούτοις, ὕστερον δὲ πειρασόμεθα λέγειν καὶ τί ἐστιν ἕκαστον καὶ διότι πᾶν μέγεθος εἰς μεγέθη διαιρετόν.
Οὐκ ἀφαιρεῖται δ᾽ ὁ λόγος οὐδὲ τοὺς μαθηματι- κοὺς τὴν θεωρίαν, ἀναιρῶν οὕτως εἶναι τὸ ἄπειρον ὥστε ἐνεργείᾳ εἶναι ἐπὶ τὴν αὔξην ὡς ἀδιεξύτητον" 80 οὐδὲ γὰρ νῦν δέονται τοῦ ἀπείρου οὐδὲ χρῶνται, ἀλλὰ μόνον εἶναι ὅσην ἂν βούλωνται τὴν πεπερα- σμένην: 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 ἔσται μεγέθεσιν.
86. ᾿Επεὶ δὲ τὰ αἴτια διήρηται τετραχῶς, φανερὸν 2088 ὅτι ὡς ὕλη τὸ ἄπειρόν ἐστιν αἴτιον, καὶ ὅτι τὸ μὲν εἶναι αὐτῷ στέρησις, τὸ δὲ καθ᾽ αὑτὸ ὑπο- κείμενον τὸ συνεχὲς καὶ αἰσθητόν. φαίνονται, δὲ πάντες καὶ οἱ ἄλλοι ὡς ὕλῃ χρώμενοι τῷ ἀπείρῳ" διὸ καὶ ἄτοπον τὸ περιέχον ποιεῖν αὐτὸ ἀλλὰ μὴ τὸ περιεχόμενον.
CHAPTER VIII ARGUMENT {A final reply ta the considerations urged (chap. iv.) in favour of the existence of something infinite.~C.]
5 Λοιπὸν δ᾽ ἐπελθεῖν καθ᾽ obs λόγους «τὸ ἄπειρον 9 [Cf Bk. IV. chap, xi. 219 a 12 (p. $85). Literally, ‘ viz. a change (is called infinite) because the magnitude over which the change—of place or quality or size—tales place (is called infinite); and time (is called infimte) because the change (it measures is so).’—C,] PHYSICS, III. vi—vut.
is so because it is the comparative register of move- ments.* For the present, however, we shall deal with them all as we want them, though we shall not forget presently to attempt to give an account of what each of them is, and why every magnitude is divisible into magnitudes.® Nor does this account of infinity rob the mathe- maticians of their study; for all that it denies is the actual existence of anything so great that you can never get to the end of it. And asa matter of fact, mathematicians never ask for or introduce an infinite magnitude; they only claim that the finite line shall be of any length they please; and it is possible to divide any magnitude whatsoever in the same pro- portion as the greatest magnitude. So that the question under discussion does not affect their de- monstrations; whereas actual dimensional existence can only be found in actually existent magnitudes.
As to the so-called ‘ four causes’ or determinants, it is obviously the ‘material determinant’ to which the unlimited must be referred, and its essence is ‘shortage,’ while the subject in which it properly inheres will be the sensible-continuous. All other thinkers, too, agree in working with the unlimited as the ‘material’ (rather than formal) determinant; hence it is absurd to regard it as that which contains rather than as that which is contained.
CHAPTER VIII Ir remains to disarm the considerations urged in + See Book IV. chaps. xi.-xiin σ [So they can obtain a line as small as they want.—C.]
ARISTOTLE 808. εἶναι δοκεῖ οὐ μόνον δυνάμει ἀλλ᾽ ὡς ddwpropevor: 1 1 2 0 5 oO τὰ μὲν γάρ ἐστιν αὐτῶν οὐκ ἀναγκαῖα, τὰ δ᾽ ἔχει τινὰς ἑτέρας ἀληθεῖς ἀπαντήσεις.
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.
Ἔστι 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.
Τὸ δὲ τῇ νοήσει πιστεύειν ἄτοπον" οὐ γὰρ ἐπὶ τοῦ πράγματος ἡ ὑπεροχὴ καὶ ἧ ἔλλειψις, ἀλλ᾽ ἐπὶ τῆς νοήσεως. ἕκαστον γὰρ ἡμῶν νοήσειέν ἄν τις πολλαπλάσιον ἑαυτοῦ αὔξων εἰς ἄπειρον" ἀλλ᾽ οὐ διὰ τοῦτο ἔξω τοῦ ἄστεός τίς ἐστιν ἢ τοῦ τηλικοῦδε μεγέθους ὃ ἔχομεν, ὅτι νοεῖ τις, ἀλλ᾽ ὅτι ἔστιν' τοῦτο δὲ συμβέβηκεν. ὁ δὲ χρόνος καὶ ® (See 203 Ὁ 18. The argument that there must be an inexhaustible reservoir (ἀρχή) from which things may come into er is ascribed to Anaximander. (ΟΝ Diels, Vors.
> [Cf 203 b 20, the argument: ‘ What is limited is always limited by coming up to something (πρός τι περαίνειν), so there can be no (absolute) limit, if one thing must always come up to another. —C.]
5 [Literally, ‘ nor is contact possible between any two things you choose to mention.’ This seems to be a distinct oint in which ‘being in contact’ differs from ‘ being imited.’—C.
4 (Cf. 208 b 22, ‘ Number is thought to be infinite, be- cause we can always conceive something beyond any limit.’~-C.]
PHYSICS, III. vu.
support of the existence of the unlimited not only as a potentiality but as actually compassed. Some of them do not follow as alleged from the admitted premises; and the rest can be met along some other line of sound reasoning.
(1) Admitting that things never cease to come into being,“ it does not follow that there actually exists some sense-perceptible body unlimited in quantity; for though the sum of things be limited, things may come out of and pass into each other without end.
(2) Again, being in contact and being limited are different things.’ Contact is a relation with some- thing else, for there must be something to touch the touched; and this may happen to something limited incidentally; but ‘ being limited’ is not a relation. Also a limited thing need not be touched by a thing homogeneous with itsclf and cannot be touched by any other.¢ (3) It is futile to trust to what we can conceive as a guide to what is or can be;# for the excess or defect in such a case lies not in the thing but in the conceiving. One might conceive any one of us to be many times as big as we are, without limit; but if there does exist a man too big for the city to hold,¢ for instance, or even bigger than the men we know of, that is not because we have conceived him to exist, but because he does; and whether we have or have not conceived him to exist is a mere incident.
Time and movement are indeed unlimited, but [ἔξω τοῦ ἄστεος should mean ‘ outside the city ’—a distinct illustration. Philoponus, 495. 6, says the most accurate copies did not contain τοῦ ἄστεος but only ἔξω τοῦ τηλικούτου μεγέθους (τουτέστιν οὗ ἔχομεν οἱ Avipwra). See Diels, Zur Textgesch. p. 39.—C.]
ARISTOTLE 2081 ἡ κίνησις ἀπειρά ἐστι, Kal ἡ νόησις, οὐχ ὑπο- μένοντος τοῦ λαμβανομένου. μέγεθος δὲ οὔτε τῇ καθαιρέσει οὔτε τῇ νοητικῇ αὐξήσει ἐστὶν ἄπειρον.
᾿Αλλὰ περὶ μὲν τοῦ ἀπείρου, πῶς ἔστι καὶ πῶς οὐκ ἔστι καὶ τί ἐστιν, εἴρηται.
« [Or, ‘Time and πιονεμπιοπὲ are infinite only as processes in which any part you take does not subsist, and it is the same with our conception of then: (z.¢., our conception can always reach out to further time and further change, but never embrace infinite time or infinite change as a completed whole); while magnitude (though infinitely divisible potentially) is not actually reduced to the infinitely small by PHYSICS, III. vin.
only as processes, and we cannot even suppose their successive stretches to coexist.
All parts of a given magnitude do indeed coexist, but neither by reduction nor by expansion can its unlimited divisibility be actually carried into execution.@ So much as to the unlimited, the sense in which it exists and does not exist, and what it is.
subtraction nor actually increased to the infinitely great by a process of augmentation that we conceive.’ These may be taken as further arguments that our unlimited power of conception cannot establish the existence of infinites other than those Aristotle has recognized.—C.]
BOOK IV INTRODUCTION Tim Greek topos, to the discussion of which the first section (Chaps. i-v.) of this Book is devoted, may mean either ‘ place’ (Latin Jocus) or ‘ space’ (Latin spatiua); but what Aristotle is directly concerned with here is only ‘ place,’ implying ‘ position,’ and not abstract or absolute “space ’ at all.
Failure to understand this has led to grotesque mis- conceptions of Aristotle’s teaching and depreciation of his intelligence.
When, in answer to the question ‘where is it δ᾽ we mention the ‘ place ’ that a thing is in, what exactly do we mean by ‘ place ’?
The question had been confused by Zeno’s attempt to show that the conception of things moving, or changing their places, or being in ‘ places’ at all, must involve us in hopeless contradictions; and Aristotle, who, unlike Zeno, believes in the reality both of time and of place, begins hig investigation, ag usual, by asking how we currently use the word, and what are the perplexities in which it seems to land us. But from the very first he keeps firm hold of the principle that would now be expressed as ‘ the relativity of position.» To speak of the ‘ absolute’ position or place—either of a point or of the universe—would have been unmeaning to him, because ‘ position ’ only exists in relation to some ‘ frame of reference.’ ¢ 4 Compare Clerk Maxwell's dictum (under the heading of “ Absolute space”) that anyone “ who will try to imagine ARISTOTLE It is impossible, then, to assign any place to the universe as a whole, and since Aristotle’s universe was bounded by the outmost heavenly sphere it was impossible to assign to that sphere (regarded as a single whole) any place at all.
But <Aristotle’s universe hag determinate dimensions, although it has no place. A thing's place, then, is not constituted by its own dimensions, nor can anything be 108 own place.
So the universe is a body, or masy, that has no place, because there is nothing else that it isim. Can there also —actually or conceivably—be a‘ place’ that ‘ hag nothing in it’?
It is obvious that this question may raise speculations as to ‘ void spaces’ between, or within, the material con- stituents of the universe, just as the ‘ unplaced ἡ universe itself suggests speculations as to an ‘ unmeasurable void ’ outside it. Thus a discussion of the ‘ vuid’ which is the subject of the second section of this Book (Chaps. vi.-ix.) naturally dovetails into the discussion of ‘ place’ in the first section. Thus it happens that suggestions, at least, of ‘ space’ or ‘ spaces’ are, so to speak, always there in the offing, though Aristotle only indirectly concerns him- aelf with them here. We may take it that κενόν [ = void comeg much nearer to our “ space” than τόπος { = place does.
We turn now to the direct examination of Aristotle’s doctrine of ‘ place.’ ‘Io the question ‘ Where is the wine?’ the answer may be ‘ In the flask’ or ‘ In the chest’ or ‘ In the house,’ and go forth, till we come ultimately to ‘ Under heaven.’
Now of these answers it is only the first that tells us of the ‘ proper’ place of the wine, which it has all to itself. The other answers refer us to places ‘common’ to the wine and to other things. But they all assign some kind of physical /imit, within which the wine is contained—the wine only, or the wine together with certain other things.
the state of a mind conscious of knowing the absolute eae of a point will ever afler be content with our relative knowledge " (Matter and Motion, London, 1876, p. 20).
PHYSICS, IV. INTRODUCTION In developing the implications of this all-important distinction between a thing’s proper place and a place which is common to it and other things, it may be helpful to vary Aristotle’s illustrations.
Suppose a stone to be sinking Lhrough a body of water (or a bubble rising through it). What is its proper place at any moment? Aristotle would answer ‘ The aqueous surface which at that moment constitutes its immediate envelope.’ This is vital to his conception of a place-proper. The placc-proper of the stone, at any moment, is not the whole body of water but the aqueous surface immediately enveloping it at that moment, and so too the proper place of the wine, as conceived by Aristotle, is not the bottle as a whole, but the inner surface of the bottle. It is there- fore (like any other surface, or length, or extension) a reality, and even a quantitive reality, for things really are long, and just so long, and really have surfaces of definite dimensionality and (m case of a spherical surface, for example) capacity. But an actual surface cannot exist apart from soine physical body whose surface it is.
Hence, for a body to be in a place it must be regarded ag itself continuous, but as differentiated from its body- continent. Undiflerentiated portions of the water in a flask, for instance, are ‘ in’ the whole mass, as the ‘ part’ ig ‘ in the whole,’ but not as the located in the location, or the content in the continent. But if a definitely limited vortex could arise within the mass it would be differ- entiated by its motion from its stationary aqueous envelope, and the surface of the latter, in contact with the vortex, would be its ‘place.’ Conceptually a continuous mass can be subdivided indefinitely, but since surfaces cannot constitute a mass any more than points can constitute a line, there must always be some continuous mass between an} two ‘ places.’ Thus a ‘place’ is never 4 Aristotle, not being an atomist, undoubtedly believed in the continuity of the ultimate matter, so that, for instance, however thin the concentric laminae into which we suppose a sphere to be divided, there must always be undivided matter between the inner and the outer surfaces of any one of them.
ARISTOTLE ‘in another place’ in the sense in which a content is in its continent.
A body, then, is in a place constituted by a surface of the body-continent 5 and this surface, though not itself a physical body, must pertain to some physical body, of which it is the smface. And that physical body (except in the case of the supreme heaven) must itself have a place. So, as we travel outwards, reaching, at every step, a place which is the proper place of the body-contment we have last reached (and the common place of it and of everything within it), we come ultimately to the inner surface of the revolying heaven, which is the universal place and ig itself unplaced. Compare Gen. Introd. p. Ixviii.