* [Or, ‘And even if it (i.e. τὸ ἔξω τοῦ οὐρανοῦ, ἄπειρον by) is (not ocenpied by other worlds, but) vacant, “place” too will then be unlimited, and there must be no limit to “body”; for (“* place” is that in which there is or may be “body ” and) in things eternal there is no distinction between possibility and actuality.’ Eudemus (Sunpl. 467. 26) quotes an argument of Archytas for infinite extension of place and body on these lines. Cf. Heath, Greek Maths. i. 214.—C,] Ὁ [Or (reading torw} ‘there nevertheless exists an ‘ in- finity ’ in the singular or the plural sense.’—C.]
4 [The classification is this: ‘ infinite’ is used to mean (a) ‘what cannot be traversed’ (διελθεῖν) at all, because it is not the sort of thing there can be any question of traversing, or (6) what can be traversed, bui ‘there is no getting to PHYSICS, III. 1.
oceupying it everywhere, for in things eternal there is no distinction between possibility and actuality? 4 But the consideration of this ‘ unlimited ’ presents us with a genuine problem; for not only by asserting but also by denying its reality we seem to be landed in a number of untenable positions. And then we have to ask whether it is a substantively existing ‘something ’; or an attribute, necessarily pertaining to some substantively existent thing in nature; or neither, though none the less unlimited, whether in its unity or its multiplicity. But the physicist’s chief concern is to inquire whether there is such a thing as an unlimited ‘ sensible ’ magnitude.
Let us begin, then, by inquiring in how many senses we use the term. If by saying that a thing has ‘no limit’ you mean ‘no boundary,’ then you mean that its nature is such that it would be nonsense to speak of ‘ passing through it from side io side,’ just in the same way that a sound is ‘ invisible.’ But you might also mean that, though it is of such nature thal you can traverse it, it does not admit (whether you are speaking absolutely or practically) of your getting ‘through’ it so as to come out beyond it. Or again you may mean that its nature would allow it to have a boundary such as to make it passable-through, but that in this case it has not.¢ Again, the ‘ absence of limit ’ may mean capacity for being multiplied indefinitely, or for being divided indefinitely, or both.
the end of it’ (τὸ διέξοδον ἔχον ἀτελεύτητον), eilher (1) in the sense that this is hardly possible or practically umpossible (ἢ ὃ μόλις), or (2) in the sense that it is absolutely impossible, because there is no actual end, though it is the sort of thing you mi ἣν ‘pet to the end of’ if it had one (ἢ ὃ πεφυκός, KTN.)—C.
τυ 18 ARISTOTLE CHAPTER V ARGUMENT [The abstract ‘ Infinity itself,’ which the Pythagoreans and Plato regarded as an element, cannot actually exist apart from sensible things (204 a 8-9). (1) If a subsiunce, it would be indivisible, but they regard it as divisible (a 9-14), If an attribute, rt cannot be an element (a 14-17). (2) There can be no ‘ infinite’ which is neither a number nor @ magnitude (a 17-20). (8) Whether this alleged Infinite is divisible or indivisible, contradictions result Physics, however, is concerned vather with the question whether an infinite material body actually exists or not Abstract arguments might be urged against its existence (b 4-10). Fram natural facts it appears that an unlimited Χωριστὸν μὲν οὖν εἶναι τὸ ἄπειρον τῶν αἰσθη- τῶν, αὐτό τι ὃν ἄπειρον, οὐχ οἷόν τε.
Ei γὰρ μήτε μέγεθός ἐστι μήτε πλῆθος, ἀλλ᾽ οὐσία αὐτό ἐστι τὸ ἄπειρον καὶ μὴ συμβεβηκός, ἀδιαίρετον gorau τὸ γὰρ διαιρετὸν ἢ μέγεθος ἔσταν ἢ πλῆθος. εἰ δὲ ἀδιαίρετον, οὐκ ἄπειρον, εἰ μὴ ὡς ἡ φωνὴ ἀόρατος’ ἀλλ᾽ οὐχ οὕτως οὔτε φασὶν εἶναι οἱ φάσκοντες εἶναι τὸ ἄπειρον οὔτε ἡμεῖς ζητοῦμεν, ἀλλ᾽’ ὡς ἀδιέξοδον. εἰ δὲ κατὰ συμβεβηκός ἐστι τὸ ἄπειρον, οὐκ ἂν εἴη στοιχεῖον 5 [The view ascribed above (208 a 4) to Plato and the Pythagoreans, who made ‘Infinity itself’ an element (σταιχεῖον) in things.—C.]
PHYSICS, TIL. v.
CHAPTER V ARGUMENT (continued) body could not be either (1) a compound of a limited number of elements (Ὁ 10-22), ov (2) a simple and uniform element, or like Anaximander’s unlimited stuff. If one element were unlimited, tt would devour the rest; and Anaaimander’s stuff does not ewist (Ὁ 22-205 a 8). A further proof, based on the doctrine that each element has a ‘ natural place’ (a 8-b 1). Anaxagoras gives the wrong reason for his ‘Unlimited’ remaining at rest (Ὁ 1-28).
An unlimited body is inconsistent with the existence of natural places (b 23-31), and of the distinctions between ‘up’ and ‘ down,’ ete., in place (Ὁ 31-85). A body must be in a place; but a place must be limited, and so no place can contain an unlimited body (Ὁ 35-206 a 7).
Thus an unlimited body cannot exrst in actuality (a 7-8).
Now, it is impossible that there should exist an ‘unlimited’ sejunct from objects of sense, and constituting a self-existing ‘ infinity.’
For (1) if the ‘ unlimited’ is neither magnitude nor number, but infinity itself is its very being and not an attribute,* it must be indivisible; for what is divisible must be either extended or numerous. But Af 1¢ is ‘ indivisible,’ it cannot be ‘ unlimited,’ except in the sense in which speech is invisible; but that is neither what those who assert the reality of the ‘unlimited’ mean by the term, nor what we are interested in investigating; for what we all mean is that which it is not possible to get through to the end of. If the unlimited exists at all, therefore, it must be attributively; but, if so, qua infinite it can no more be a constituent element of substantively ARISTOTLE ARISTOTLE 204aTav ὄντων, ἧ ἄπειρον, ὥσπερ οὐδὲ τὸ ἀόρατον τῆς διαλέκτου, καΐτοι ἡ φωνή ἐστιν ἀόρατος. "Eat πῶς ἐνδέχεται εἷναί τι αὐτὸ ἄπειρον, εἴπερ μὴ καὶ ἀριθμὸν καὶ μέγεθος, ὧν ἐστι καθ᾽ αὑτὸ πάθος τι τὸ ἄπειρον; ἔτι γὰρ ἧττον ἀνάγκη ἢ é 29 τὸν ἀριθμὸν ἢ τὸ μέγεθος. 3 Φανερὸν δὲ καὶ ὅτι οὐκ ἐνδέχεται εἶναι τὸ ἄπειρον ὡς ἐνεργείᾳ ὃν καὶ ὡς οὐσίαν καὶ ἀρχήν. lot a ι ἔστων γὰρ ὁτιοῦν αὐτοῦ ἄπειρον τὸ λαμβανόμενον, εἰ μεριστόν--τὸ γὰρ ἀπείρῳ εἶναι καὶ ἄπειρον τὸ αὐτό, εἴπερ οὐσία τὸ ἄπειρον καὶ μὴ καθ᾽ ὑποκει- 25 μένου--ὥστ᾽ ἢ ἀδιαίρετον ἢ εἰς ἄπειρα διαιρετόν, πολλὰ δ᾽ ἄπειρα τὸ αὐτὸ εἶναι ἀδύνατον: ἀλλὰ μὴν ὥσπερ ἀέρος ἀὴρ μέρος, οὕτω καὶ ἄπειρον ἀπείρου, εἴ ye οὐσία ἐστὶ καὶ ἀρχή. ἀμέριστον dpa καὶ ἀδιαίρετον. ἀλλ᾽ ἀδύνατον τὸ ἐντελεχείᾳ ὃν ἄπειρον (ποσὸν γάρ τι εἶναι ἀναγκαῖον). κατὰ 2 80 συμβεβηκὸς ἄρα ὑπάρχει τὸ ἄπειρον. ἀλλ᾽ εἰ οὕτως, εἴρηται ὅτι οὐκ ἐνδέχεται αὐτὸ λέγειν ἀρχήν, ἀλλ᾽ ἐκεῖνο ᾧ συμβέβηκεν--τὸν ἀέρα ἢ τὸ ἄρτιον. ὥστε ἀτόπως ἂν ἀποφαίνοιντο οἵ , we ν ε,, av λέγοντες οὕτως ὥσπερ of Tlvfaydperot dacw: ἅμα f a γὰρ οὐσίαν ποιοῦσι τὸ ἄπειρον καὶ μερίζουσιν. AV ἴσως αὕτη μὲν ἔστι καθόλου ἡ ζήτησις a [Literally y, ‘“to be infinite” (infiniteness) and “‘ the infinite ” are the same thing.’ 115 infiniteness is ex hypothesi the whole of its nature; so, if divided, its parts would be essentially and by nature infinite—C, ] > (Literally, ‘ But an actually existing infinite could not ων one into parts), for it would have to be a quantity.’
PHYSICS, IIT. v.
existent things than ‘ invisibility ’can be asubstantive component of ‘ language,’ although sound is invisible.
(2) Again, how can the unlimited possibly have a substantive existence if magnitude and number themselves, of which it is, by its very nature. an affection or qualification, have no such existence? The substantive existence of the unlimited must be even more impossible than that of magnitude and number, (3) It is further manifest that infinity cannot exist as an actualized entity and as substance or principle. For in that case, if it were divisible, then any several portion of it that you might take would itself be ‘infinity "—for, if ‘ infinity ’ is a substance and not an attribute, then any portion of that which is infinity’s self must by definition be infinite?—so that infinity must be either (a) altogether indivisible or (b) divisible into infinites. But (6) for one same thing to be many infinites is impossible; and yet this would follow, if infinity had a substantive existence and were an element, just as a portion of air is itself air. (a) If a substance, then, it must be indivisible and must not contain parts homogeneous with itself. But in that case it cannot be a realized entity at all; for if so, being essentially a quantum, it would be divisible. Infinity, then, must exist, if at all, attributively. But in that case it has been shown that it cannot be infinity’s self that is an element, but the infinite thing of which infinity is an attribute, as some have predicated of ‘ air’ or of ‘the even number.’ This shows the futility of the Pythagorean contention, which gives infinity itself a substantive existence, and yet makes it divisible.
But, after all, the general question of the possibility ARISTOTLE 804 4 86 μᾶλλον, εἰ ἐνδέχεται τὸ ἄπειρον Kal ἐν τοῖς μαθη- 804} ματικοῖς εἶναι καὶ ἐν τοῖς νοητοῖς καὶ μηδὲν ἔχουσι μέγεθος" ἡμεῖς ὃ ἐπισκοποῦμεν περὶ τῶν αἰσθητῶν καὶ περὶ ὧν ποιούμεθα τὴν μέθοδον, ἄρ᾽ ἔστιν ἐν αὐτοῖς ἢ obi ἔστι σῶμα ἄπειρον ἐπὶ τὴν αὔξησιν.
5 Δογικῶς μὲν οὖν σκοπουμένοις ex 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 ἀρι- 10 θμητὸν ἐνδέχεται ἀριθμῆσαι, καὶ διεξελθεῖν ἂν εἴη δυνατὸν τὸ ἀπειρον.
Φυσικῶς δὲ μᾶλλον θεωροῦσιν ἐκ τῶνδε: οὔτε γὰρ σύνθετον οἷόν τε εἶναι οὔτε ἁπλοῦν. Σύνθετον μὲν οὖν οὐκ ἔσταν τὸ ἄπειρον σῶμα, εἰ πεπερασμέμα τῷ πλήθει τὰ στοιχεῖα. ἀνάγκη γὰρ πλείω εἶναι, καὶ ἰσάζειν ἀεὶ τἀναντία, καὶ μὴ τ6 εἶναι ἕν αὐτῶν ἄπειρον: εἰ γὰρ ὁποσῳοῦν λείπεται ἡ ἐν ἑνὶ σώματι δύναμις θατέρου, οἷον εἰ τὸ πῦρ πεπέρανται ὁ δ᾽ ἀὴρ ἄπειρος, ἔστι δὲ τὸ ἴσον πῦρ τοῦ ἴσου ἀέρος τῇ δυνάμει ὁποσαπλασιονοῦν, μόνον δὲ ἀριθμόν τινα ἔχον, ὅμως φανερὸν ὅτι τὸ 5 [ἐπὶ τὴν αὔξησιν, as opposed to infinite divisibility.—C.] > [NoyixGs, as here opposed to φυσικῶς, Means arguing from abstract principles (λόγοι) rather than from the concrete facts (¢pya) of our science.—C.]
5 See Gen. Introd. pp. Ixxxin f.
4 [Literally, ‘Nor yet (can there exist) a number con- ceived as existin separately (from numbered things) and as infinite.’ ‘The infinite’ cannol exist by itself in the form of an infinite number, any more than as an infinite bedy.—C.]
® [Otherwise the body would not be a compound,—C.]
PHYSICS, III. v.
of the unlimited in mathematics and, in the concep- tual sphere, in things that have no magnitude, lies beyond our present scope; for we are engaged in the study of things cognizable by the senses, and what we are concerned to know is whether there exists any actual physical body or substance that is unlimited in expansion.* If we argue on abstract grounds,’ the following considerations would appear to be conclusive against the existence of any such body. If ‘body’ is defined as ‘that, the limit or boundary of which is surface,’ ¢ an unlimited body is impossible, either conceptually or sensibly. Nor can ‘a number,’ even if abstract, be infinite; ὦ for both numbers themselves and the things that are numbered are ‘ numerable,’ i.e., can be counted. If, then, ‘a number’ is by definition counLable, then, if it were also unlimited, the un- limited would be passed through to the end or limit, which is a contradiction, If, on the other hand, we argue from the natural properties of things, we may add these further considerations: the supposed infinite body can be neither (1) compound nor (2) simple.
(1) It cannot be compound, if the elemental sub- stances are limited in number. For these elements must be more than one,’ and they must permanently balance each other’s contrasted qualities, so that no one of them must be infinite. For however relatively feeble the power of one element (say, air) might be with respect to that of another (say, fire), yet, if fire were limited, the air unlimited in quantity, and if there were any ratio of equivalence whatever between the assimilative power of a given volume of air and that of an equal volume of fire, then obyiously ARISTOTLE 804» ἄπειρον ὑπερβαλεῖ καὶ φθερεῖ τὸ πεπερασμένον. 20 ἕκαστον δ᾽ ἄπειρον εἶναι ἀδύνατον: σῶμα μὲν γάρ ἐστι τὸ πάντῃ ἔχον διάστασιν, ἄπειρον δὲ τὸ ἀπεράντως διεστηκός, ὥστε τὸ ἄπειρον σῶμα πανταχῇ ἔσται διεστηκὸς εἰς ἄπειρον.
"AMG μὴν οὐδὲ ἕν καὶ ἁπλοῦν εἶναι ἐνδέχεται τὸ ἄπειρον σῶμα, οὔτε ὡς λέγουσί τινες τὸ παρὰ τὰ στοιχεῖα, ἐξ οὗ ταῦτα γεννῶσιν, οὔθ᾽ ἁπλῶς.
85 εἰσὶ γάρ τινες οἱ τοῦτο ποιοῦσι τὸ ἄπειρον ἀλλ᾽ οὐκ ἀέρα ἢ ὕδωρ, ὡς μὴ τἄλλα φθείρηται ὑπὸ τοῦ ἀπείρου αὐτῶν' ἔχουσι γὰρ πρὸς ἄλληλα ἐναντίωσιν —olov 5 μὲν ἀὴρ ψυχρός, τὸ δ᾽ ὕδωρ ὑγρόν, τὸ δὲ πῦρ θερμόν--ὧν εἰ ἣν ἕν ἄπειρον, ἔφθαρτο ἂν ἤδη τἄλλα" νῦν δ᾽ ἕτερον εἶναί φασιν ἐξ οὗ ταῦτα.
a0 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 ἔστι τοιοῦτον σῶμα αἰσθητὸν παρὰ τὰ στοιχεῖα καλούμενα: ἅπαντα γὰρ ἐξ οὗ ἐστι, καὶ διαλύεται εἰς τοῦτο, ὥστε ἦν ἂν ἐνταῦθα παρὰ 80 ἀέρα καὶ πῦρ καὶ γῆν καὶ ὕδωρ' φαζεται δ᾽ οὐδέν. 2058 οὐδὲ δὴ πῦρ οὐδ᾽ ἄλλο τι τῶν στοιχείων οὐδὲν ἄπειρον ἐνδέχεται εἶναι. ὅλως γὰρ καὶ χωρὶς τοῦ > [Anaximander held that his infinite stuff existed in its undifferentiated state outside our world, which it envelopes. Ye also explicitly asserted ἐξ ὧν ἡ γένεσίς ἐστι τοῖς οὖσι, καὶ τὴν φϑορὰν εἰς ταῦτα γἔγνεσθαι, Diels, Vors, 2, 9.---Ο,] PHYSICS, III. v the unlimited volume of air must vauquish and destroy the limited volume of fire. Nor could every element severally be unlimited; for ‘ body ’ is that which has dimension in every direction, and the unlimited that which extends without limit; so an unlimited body must extend without limit in every direction.
(2) But if the infinite body cannot be compound, neither can it be simple and uniform, either (a) as itself the universal clement, or (6) (as some ἃ have supposed) as an unlimited something in addition to the elements, the matrix out of which they come. (b) The reason for supposing this additional something, rather than air or water for instance, to be the unlimited, was that it seemed to evade the dilemma set out above. For since air is cold, water moist, and fire hot, and these properties are mutually destructive, an infinity of one of them would mean that the others would have perished by this time; but this, they say, would not apply to an undifferentiated something out of which they all came. But no such thing can exist; not because of its infiniteness (as to which a more general consideration, equally applicable to air or water or any other substance whatever, will be urged in the immediate sequel), but because the senses reveal no such body (by hypothesis ‘ sensible’) in addition to the elements as commonly enumerated; for whatever things come out of when they come into existence, that they return into when they pass out of existence, so that if the elements came out of something else, that something would exist in our world ® over and above air, fire, earth, and water; but no such thing is apparent to observation. (a) Nor can fire or any other of the elements possibly be unlimited. For— on 10 ARISTOTLE on 10 ARISTOTLE ΝΜ εν a » κα >, A fal at ἄπειρον εἶναί τι αὐτῶν, ἀδύνατον τὸ πᾶν, κἂν 7 πεπερασμένον, ἢ εἶναι ἢ γίγνεσθαι ἕν τι αὐτῶν, ὥσπερ ᾿ΗἩράκλειτός φησιν ἅπαντα γίνεσθαί ποτε πῦρ. ὃ 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. δεῖ δὲ περὶ παντὸς καὶ ἐκ τῶνδε' σκοπεῖν, εἰ ἐνδέχεται ἢ οὐκ ἐνδέχεται εἶναι ἄπειρον."
Ὅτι δ᾽ ὅλως ἀδύνατον εἶναι σῶμα ἄπειρον αἰσθητόν, ἐκ τῶνδε δῆλον.
Πέφυκε γὰρ πᾶν τὸ αἰσθητὸν ποὺ εἶναι, καὶ ἔστι τόπος τις ἑκάστου, καὶ ὁ αὐτὸς τοῦ μορίου καὶ παντός, οἷον ὅλης τε τῆς γῆς καὶ βώλου μιᾶς, a tA 2 kal πυρὸς καὶ σπινθῆρος. ὥστε εἰ μὲν aie ἀκίνητον ἔσται ἢ ἀεὶ οἰσθήσεται. καίτοι ἀδύνατον" τί γὰρ μᾶλλον κάτω ἢ ἄνω ἢ ὁπουοῦν"; λέγω ὃ 1 (To avoid ake this sentence a mere duplicate of the next, I have followed Philoponus (436. 1). περὶ (I, cory. F, κατὰ ἘΠ) παντὸς καὶ (ΕἸ: E omits καὶ) ἐκς τῶν δε Simplic. 482. 1 (lemma), Philop. 4136, 1 (lemma).—C,] a jee ἄπειρον Simplic, 482.6 (paraphy.): εἶναι ἀρχήν (ἄπειρον Ὁ) Philop. 436, 6(paraphy.): εἶναι i: εἶναι σῶμα (σῶμα εἶναι 1) ἄπειρον αἰσθητὸν cett.—C.]
* [See frag. 63-66. It is disputed whether Heracleitus really meant a general conflagration (which seems in- consistent with his system), or that every part of the universe at some time passes through the fiery phase in the cycle of transformation.—C.]
» (And if the universe were one element or Anaximander’s undifferentiated unity, there would be no opposite, and hence change would be impossible.—C.]
Ὁ Fe question set aside above (1. 2).—C.] 4 [Aristotle assumes his own doctrine that each element has 8. natural region (the centre for earth, the circumference PHYSICS, III. v.
more generally and quite apart from the question of its being limited or unlimited—it is impossible that all the universe, even if it be limited, should be or should become any single one of the elements, as Heracleitus* supposed, when he said that all things soinetime become fire. Against this the same argument can be urged as applies to the undifferentia~ ted unity or matrix such as some physicists have assumed in addition to the elements; for things always change from one term of an opposition to the other, from hot to cold, for instance.? But the question whether an element can or cannot be unlimited ¢ should be considered in the case of every element (not only of fire) in the light of the ἡ following considerations also.
So far on the supposition that the elemental bodies are limited in number; but as to the more general question whether it 1s possible in any case that a sensible body should be without limit, the following considerations are conclusive against any such possibility.
A sensible body or substance must be situated somewhere, and any given body has some proper site, which is the same for the part as for the whole; for instance, for earth as a whole and for a single clod of earth, or for fire as a whole and for a single spark? Consequently (1) if the whole infinite body * were of uniform substance, it would either be motionless or always in motion: But the latter is impossible; for why more down than up, or to anywhere distinguishable from anywhere else? I for fire, etc.), to which it is of its very nature to move, or in which it is of rts nature to rest, De caslo, i, 2.—C.] * (Se. τὸ πᾶν (as in J, 19), the ‘ whole’ opposed to the * part’ (as in ἐς 11).—C.]
ARISTOTLE nt a / a 206.15 οἷον εἰ βῶλος εἴη, ποῦ αὕτη κινηθήσεται ἢ ποῦ μενεῖ; 6 yap τόπος ἄπειρος τοῦ συγγενοῦς αὐτῇ σωμαᾶτος. πότερον οὺν Ka έξει τον OAOV ΤΟΤΟῸΡ; καὶ πῶς; Tis οὖν ἢ ποῦ ἡ μονὴ καὶ ἡ κίνησις JOA δ n~ ~ 3 On Μ αὐτῆς; ἢ πανταχοῦ μενεῖ (οὐ κινηθήσεται ἄρα) ἢ πανταχοῦ κινηθήσεται (οὐκ ἄρα στήσεται). εἰ lod " 80 δ᾽ ἀνόμοιον τὸ πᾶν, ἀνόμοιοι καὶ οἱ τόποι" καὶ πρῶτον μὲν οὐχ ἕν τὸ σῶμα τοῦ παντὸς ἀλλ᾽ ἣ τῷ ἅπτεσθαι: ἔπειτα ἤτοι πεπερασμένα ταῦτ Η n \ ων ἔσται ἢ ἄπειρα τῷ εἴδει. πεπερασμένα μὲν οὖν 3 Ye οὐχ οἷόν τε' ἔσται yap τὰ μὲν ἄπειρα τὰ δ᾽ οὔ (εἰ τὸ πᾶν ἄπειρον), οἷον τὸ πῦρ ἢ τὸ ὕδωρ, φθορὰ BY 5 fa) a yt tf / wv ε δὲ τὸ τοιοῦτον τοῖς ἐναντίοις, καθάπερ εἴρηται 7 πρότερον. (καὶ διὰ τοῦτ᾽ οὐθεὶς τὸ ἕν καὶ ἄπειρον πῦρ ἐποίησεν οὐδὲ γῆν τῶν φυσιολόγων, ἀλλ᾽ ἣ ὕδωρ ἢ ἀέρα ἣ τὸ μέσον αὐτῶν, ὅτι τόπος ἑκατέρου δῆλος ἣν διωρισμένος, ταῦτα δ᾽ ἐπαμφοτερίζει τῷ 0 ἄνω καὶ κάτω) εἰ δ᾽ ἄπειρα καὶ ἁπλᾶ, καὶ of ΤΆ af A μὲ u x ~ > τόποι ἄπειροι καὶ ἔσται ἄπειρα τὰ στοιχεῖα. εἰ δὲ nt ἡδύ ᾿, ς 4 € τοῦτ ἀδύνατον καὶ πεπερασμένοι of τόποι, καὶ pe Ξα τς τι οι ες «᾿ς PHYSICS, ΠῚ, v.
mean: Suppose there is a clod. Whither is it to move, or where to rest, through the infinite region of substance indistinguishable from itself? Will it take possession of it all? But how can it? What then and where is its rest, or what and whither its movement? Hither it will be at rest everywhere, in which case it will never move; or it will every- where be in motion, and then it will never come to rest. If (2), on the other hand, the universe is not all homogeneous, then the sites proper to its unlike constituents are distinguishable; and in the first place the universe will have no unity save that of continuons contact of part with part; and in the second place the differences in ind betwecn the parts will be either (a) limited or (5) unlimited in number. (a) They cannot be limited; for in that case, if the universe itself were unlimited in quantity, some con- stituents of it would have to be unlimited in quantity, others limited (say, fire limited and water unlimited), and the unlimited element would have destroyed its opposite and reduced the universe to uniformity, as we said above.* (This is why no one of the physicists has taken fire or earth as his unlimited unity, but water or air or something between them has always been selected; fire and earth have each a distinct region about which there could be no mistake, whereas water and air appear to be susceptible to move- ment either up ordown.) (6) If, on the contrary, there were no limit to the differences of kind between the parts, and each part were a primary body, we should have an unlimited number of elements, and there would be no limit to the number of sites natural to them. But if there cannot be an unlimited number of elements and the sites are limited in number, the ARISTOTLE a 3 ta ‘ x 205a7d ὅλον πεπεράνθαι ἀναγκαῖον. ἀδύνατον γὰρ μὴ ἀπαρτίζειν τὸν τόπον καὶ τὸ σῶμα’ οὔτε γὰρ 6 τόπος ὁ πᾶς μείζων ἢ ὅσον ἐνδέχεται τὸ σῶμα as ἅμα εἶναι (ἅμα δ᾽ οὐδ᾽ ἄπειρον ἔσται τὸ σῶμα), - a bol 7 aN \ ‘ mu 205b οὔτε τὸ σῶμα μεῖζον ἢ ὁ τόπος" ἢ yap κενὸν ἔσται τι, ἢ σῶμα οὐδαμοῦ πεφυκὸς εἶναι. ᾿Αναξαγόρας δ᾽ ἀτόπως λέγει περὶ τῆς τοῦ ἀπείρου μονῆς" στηρίζειν γὰρ αὐτὸ αὑτό φησι τὸ » n ΄ a Ly ¢ A” ἄλλ, ‘ ἡδὲ ἄπειρον" τοῦτο δέ, ὅτι ἐν αὑτῷ (ἄλλο γὰρ οὐδὲν περιέχει), ὡς ὅπου av τι ἧ, πεφυκὸς ἐνταῦθα ef- vat. τοῦτο δ᾽ οὐκ ἀληθές" εἴη γὰρ dv τί που βιᾷ καὶ οὐχ οὗ πέφυκεν. εἰ οὖν ὅτι μάλιστα μὴ κινεῖται τὸ ὅλον (τὸ γὰρ αὑτῷ στηριζόμενον καὶ ἐν αὑτῷ ov ἀκίνητον εἶναι ἀνάγκη), ἀλλὰ διὰ τί οὐ πέφυκε κινεῖσθαι, λεκτέον: οὐ γὰρ ἱκανὸν τὸ οὕτως εἰπόντα ἀπηλλάχθαι. εἴη γὰρ ἂν καὶ ὁτιοῦν 10 ἄλλο οὐ κινούμενον, ἀλλὰ πεφυκέναι οὐδὲν κωλύει. ἐπεὶ καὶ ἡ γῆ od φέρεται, οὐδ᾽ εἰ ἄπειρος ἦν, εἰργμένη μέντοι ὑπὸ τοῦ pécou' GAN’ οὐχ ὅτι οὐκ ὃ'Ἕ᾽ δ a ἔστιν ἄλλο οὗ ἐνεχθήσεται μείνειεν dv ἐπὶ τοῦ μέσου, ἀλλ᾽ ὅτι πέφυκεν οὕτως" καίτοι ἐξείη ἂν cd ~ λέγειν ὅτι στηρίζει αὑτήν. εἰ οὖν μηδ᾽ ἐπὶ τῆς ~ a t ιό γῆς τοῦτο αἴτιον ἀπείρου οὔσης, ἀλλ᾽ ὅτι βάρος ἔχει, τὸ δὲ βαρὺ μένει ἐπὶ τοῦ μέσου, ἡ δὲ γῆ + [ὑπὸ Simpl. 486. 10 (paraphrase), Bonitz: ἀπὸ Philop. 151.
® [That there cannot be an unlimited number of elements was proved in i. 6, and it follows that the number of natural sites must be limited, and so their sum will be hmited.—C.]
PHYSICS, Ill. v.
whole must also be limited. For we cannot suppose that ‘ localily ’ in general and ‘ body ’ in general? do not fit; since neither can the totality of ‘ place’ have a greater magnitude than the totality of ‘ body’ can have at the same time—and besides, if this were so, body would not be infinite ~—noyr can ‘body ’ be in excess of ‘ place’; otherwise, in the one case there would be vacant place, and in the other body of such a nature as to have no locality.
The remarks of Anaxagoras as to the Unlimited being stationary are futile; for he says that the Un- limited stabilizes itself because it is self-contained (smmce nothing embraces it)—as if it followed thal, wherever a thing is, that must be its natural site. But this is not true, for a thing might be forced out of ils natural site. However true it may be, therefore, that the universe does not move (for that which is self- stabilized and contained in itself must be motionless), it remains to be shown why it 1s its nature so to be 3 it will not do to let Anaxagoras off on his mere asserlion. For any other body might be in fact motionless, and yet be naturally capable of motion. The earth is in fact motionless, and would be, if limitless, provided it were still held by the centre; but the reason why it would stay at the centre is not that it would have nowhere else to go, but that such is its nature; yet you might just as well say that the earth ‘ stabilizes itself’ as that the Unlimited does so. If, then, in the case of the earth (supposed to be unlimited) this would not be why it was at the centre, but because it is heavy and what is heavy remains υ [τὸν τόπον καὶ τὸ σῶμα could also mean ‘ the place (of a body) and the body (occupying that place).’—C.]
° [Because place would be larger, and there cannot be 8 magnitude larger than an infinite magnitude.—C. ] you, I R 241 ARISTOTLE 205b ἐπὶ τοῦ μέσου, ὁμοίως ἂν καὶ τὸ ἄπειρον μένοι ἐν αὑτῷ διά τιν᾽ ἄλλην αἰτίαν καὶ οὐχ ὅτι ἄπειρον καὶ στηρίζει αὐτὸ αὑτό. ἅμα δὲ δῆλον ὅτι κἂν ὁτιοῦν μέρος δέοι μένειν: ὡς γὰρ τὸ ἄπειρον ἐν 20 ἑαυτῷ μένει στηρίζον, οὕτω 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 ἄνωλ' ὥστε εἰ τοῦ ἀπείρου τόπος τὸ ἐν αὑτῷ, καὶ τοῦ μέρους 6 αὐτός' μενεῖ ἄρα ἐν ἑαυτῷ. 8. Ὅλως δὲ φανερὸν ὅτι ἀδύνατον ἅμα ἄπειρον λέγειν σῶμα καὶ τόπον τινὰ εἶναι τοῖς σώμασιν, εἰ πᾶν σῶμα αἰσθητὸν ἢ βάρος ἔχει. ἣ κουφότητα, καὶ εἰ μὲν βαρύ, ἐπὶ τὸ μέσον ἔχει τὴν ορὰν φύσει, εἰ δὲ κοῦφον, ἄνω" ἀνάγκη γὰρ καὶ τὸ ἄπειρον, ἀδύνατον δὲ ἣ ἅπαν ὁποτερονοῦν ἢ τὸ 80 ἥμισυ ἑκάτερον πεπονθέναι" πῶς γὰρ διελεῖς; ἢ πῶς τοῦ ἀπείρου ἔσται τὸ μὲν ἄνω τὸ δὲ κάτω, ἣ ἔσχατον ἢ μέσον; Ἔτι πᾶν σῶμα αἰσθητὸν ἐν τόπῳ, τόπου δὲ εἴδη καὶ διαφοραὲ τἄνω καὶ κάτω καὶ ἔμπροσθεν καὶ ὄπισθεν καὶ δεξιὸν καὶ ἀριστερόν' καὶ ταῦτα οὐ μόνον πρὸς ἡμᾶς καὶ θέσει, ἀλλὰ καὶ ἐν αὐτῷ 855 τῷ ὅλῳ διώρισται. ἀδύνατον δ᾽ ἐν τῷ ἀπείρῳ εἶναι ταῦτα.
206 a ᾿Απλῶς δ᾽ εἰ ἀδύνατον τόπον ἄπειρον εἶναι, ἐν τόπῳ δὲ πᾶν σῶμα, ἀδύνατον ἄπειρόν τι εἶναι σῶμα. ἀλλὰ μὴν τό γε ποὺ ἐν τόπῳ, καὶ τὸ ἐν ® [And consequently there will be no motion at all; but Anaxagoras did not hold that there was no motion.—C. ] » See Gen. Introd. p. Ixii.
PHYSICS, III. v.
al the centre, and the earth does so, similarly, if the Unlimited abides ‘ in itself,’ it may be for some other reason than that it is unlimited and self- stabilized. And, moreover, if it is the nature of the unlimited element, gua unlimited, to abide where it is, every particle of it ought so to abide, and therefore every particle to abide in itself. For the localities of the whole and the part are homogeneous (as ‘below’ for earth at large and any one clod, or ‘above ’ for fire at large or any one spark); hence, if the locality of the unlimited 1s ‘ in itself,’ so likewise must the locality of any part of it be. It will there- fore remain ‘ in itself,’ ἃ But in truth it is antecedently impossible that there should be an unlimited body and yet that the different substances should each have a proper locality, if we grant that every sensible body has either weight or lightness, and, if heavy, tends downwards and, if light, upwards; for the unlimited body, too, must conform to this condition, and it is impossible that it should be either heavy or light in its entirety, or half one and half the other. For how could you determine the division? Or how could there be an up or down or an inner and outer in the unhmited?
Yet again, every sensible body has locality, and the distinctions between localities are ‘ above and below,’ ‘before and behind,’ and ‘to right and to left’; and these distinctions are not merely relative to us or conventional, but are defined in the universe itself.? Yet such distinctions cannot exist in the unlimited.
In a word, if there can be no such thing as an unbounded locality, and if every body has its locality, it is impossible that a body should be unlimited. Now whatever is ‘somewhere’ is in a place; and ARISTOTLE 206 τόπῳ mov. εἰ οὖν μηδὲ ποσὸν οἷόν τ᾽ εἶναι τὸ ἄπειρον--ποσὸν γάρ τι ἔσται, οἷον δίπηχυ ἢ τρίπηχυ: ταῦτα γὰρ σημαίνει τὸ ποσόν--οὕτω 5 καὶ TO ἐν TOT, ΟΤὺ TOU’ τοῦτο δὲ μ] ἄνω 7) κατ 7 ev ἄλλῃ τινὶ διαστάσει τῶν ἕξ’ τούτων δ᾽ ἕκαστον πέρας τί ἐστιν.
Ὅτι μὲν οὖν ἐνεργείᾳ οὐκ ἔστι σῶμα ἄπειρον, φανερὸν ἐκ τούτων.
CHAPTER VI ARGUMENT [But infinity must somehow exist; otherwise time would have a beginning and end, there would be indivisible magni- tudes, and counting would come to an end (206 a 9-14), The divisibility of spatial magnitude is potentially un- limited; but this potentiality will never be realized by actual division into an unlimited number of parts. The infinite only ewists in the same sense that a day ewists, by continu- ously coming into being; but in spatial magnitude cach part that is talen persists, whereas in time and the succession of generations it is superseded by the next (a 14-b 8).
dna sense the unlimited by way of addition is the same as the unlimited by way of division. The subtraction of equal Ὅτι δ᾽, εἰ μὴ ἔστιν ἄπειρον ἁπλῶς, πολλὰ + [E omits τὸ ἐν, Bonitz conjectured οὕτως οὐδὲ ἐν τόπῳ.--- " s { have understood οὕτω καὶ τὸ ἐν τόπῳ (σημαίνει), ὅτι wot, leaving the construction of the whole sentence rather irregular. Cf. Met. 1067 a 81 τὸ γὰρ ἐν τόπῳ πού, Τοῦτο δὲ σημαίνει ἢ ἄνω ari\.—C.]
PHYSICS, II. v.—vt.
whatever is in a place is ‘somewhere.’ So if the un- limited cannot be a quantum at all, because, ifit were, it must be of some defimte quantity, such as two or three cubits length, these definite quantities being what Quantity denotes; in the same way being ina place means being ‘ somewhere,’ ὦ and that means either above or below or in some other of the six directions; but each of these is a limit.?
From all these considerations it is evident that an unlimited body cannot exist in accomplished fact.
CHAPTER VI ARGUMENT (continued) parts from a finite whole will finally exhaust the whole; but the subtraction of parts diminishing in @ constant ratro will not. Conversely the addition of parts diminishing in a constant ratio will never construct a finite whole. In this sense a limited magnitude may be the actual seat of a potentially unlimited process (Ὁ 3-33), The infinite is ‘that of which there is always more beyond’; ‘that beyond which there is nothing’ 18 the ‘whole. The unlimited is analogous to the * material’ factor from which the ‘whole’ is constituted by the ‘formal’ limiting) factor, and 80 cannot ‘embrace’ anything Yer, if we frankly deny that anything at all can be without limit, we commit ourselves to many state- » (Hence, if an unlimited body were in a place, it would be limited (which 1s a contradiction); if not in a place, it cannot be a body.—C.] 5 ARISTOTLE 206010 ἀδύνατα συμβαίνει, δῆλον: τοῦ τε yap χρόνου ἔσται τις ἀρχὴ καὶ τελευτή, καὶ τὰ μεγέθη οὐ διαιρετὰ εις μεγέθη, καὶ ἀριθμὸς οὐκ ἐεσται ἄπειρος. ὅταν δὲ διωρισμένων οὕτως μηδετέρως φαίνηται ἐνδέχεσθαι, διαιτητοῦ δεῖ, καὶ δῆλον ὅτι πὼς μὲν ἔστι πὼς δ᾽ οὔ. Λέγεται δὴ τὸ εἶναι τὸ μὲν δυνάμει τὸ δὲ ἐνii τελεχείᾳ, καὶ τὸ ἄπειρον ἔστι μὲν προσθέσει ἔστι δὲ καὶ ἀφαιρέσει. τὸ δὲ μέγεθος ὅτι μὲν κατ᾽ ἐνέργειαν οὐκ ἔστιν ἄπειρον, εἴρηται" διαιρέσει δ᾽ ἐστίν (οὐ yap χαλεπὸν ἀνελεῖν Tas ἀτόμους γραμ.- bY pds): λείπεται οὖν δυνάμει εἶναι τὸ ἄπειρον. οὐ δεῖ δὲ τὸ δυνάμει ὃν λαμβάνειν ὥσπερ εἰ δυνατὸν 90 τοῦτ᾽ ἀνδριάντα εἶναι, ὡς καὶ ἔσται τοῦτ᾽ ἀνδριάς, οὕτω καὶ ἄπειρόν τι ὃ ἔσται ἐνεργείᾳ" ἀλλ᾽ ἐπεὶ πολλαχῶς τὸ εἶναι, ὥσπερ ἡ ἡμέρα ἔστι καὶ ὁ ἀγὼν τῷ ἀεὶ ἄλλο καὶ ἄλλο γίνεσθαι, οὕτω καὶ TO ἄπειρον. καὶ yap ἐπὶ τούτων ἐστὶ καὶ δυνάμει 2 [The doctrine of indivisible lines, ascribed to Plato and his successor Xenocrates, is discussed in Book Vf, chaps, iii, See also Gen. Introd. Pp. Ixxxit f., Ixxvix note a, and Mr, Ross on Met, 992 a 20.—C.]
PHYSICS, ΠῚ, νι.
ments that are obviously false; for we should have to say that time had a beginning and will have an end, and that there are magnitudes that cannot be divided into magnitudes. and that numeration has a limit. So if we conceive the alternative to lie between the existence of some unlimited substance cognizable by the senses and there being nothing unlimited at all, we shall be landed in impossibilities either way; so we must appeal to an umpire, who will obviously have to explain that in one sense the unlimited exists and in another sense not.
Now things may be said to exist as potentialities or as actualities; and there may be an unlimited potentiality of additional terms to a convergent series, whether by addition or subtraction; and though we have seen that a magnitude cannot actually be increased beyond limit by multiplication, it may be divided into something smaller yet than any parvitude you choose to mention—for there is no difficulty in refuting the doctrine that there are such things as atomic lines.? It results that the unlimited exists as potentiality. But how are we to understand ‘ potentiality’ here? Not in the sense in which we say that the potentiality of the statue exists in the bronze; for that implies that the whole of the bronze may actually become the statue, whereas it is not so with an unlimited potentiality, since it can never become an unlimited actuality, As to this we must not be misled by the ambiguity of the word ‘is,’ for the only sense in which the unlimited is actualized at all is the sense in which we say that it “actually is’ such and such a day of the month, or that the games ‘actually are’ on; for in these cases, too, the period of time or the succession of ARISTOTLE