potentiality, as long as it is such, is by its nature uncompleted, and therefore its actual functioning —which motion is—must stop short of the comple- tion: on the attainment of the end, the motion towards it no longer exists, but is merged in the reality. Hence the difficulty of seizing its natme. For what can it be, one may ask, save a shortage, or a potentiality, or a full attainment? But none of these seems possible. It remains then, as has been said, to define it as a kind of realization or attainment, but the kind we have deseribed, difftcult to pin down, it is true, but not impossible to be.
Now? everything that is capable of molion and which is at rest when not moving, is itself in motion whenever it produces motion in anything else. For what we mean by ‘station’ or ‘ being ai rest’ is the absence of motion in that to which motion is possible, and its actualizing of this specific potentiality is just that we mean by the very act of moving. But this is effected by the body which has the potentiality of being in motion receiving the impact of a bod which is in actual motion, so that the body that becomes active, by entering into motion, at the same time becomes passive to the impact that keeps it in motion; but this is incidental, for primarily the movement is the realization of the thing’s capacity for being in motion.?
but are not affected by 1t; whereas bodies in the sublunary world have reciprocal contact, which involves that they alter and are altered by one another. The Heavens are κινοῦντα, but not δυνάμει κινητὰ καὶ ὧν ἡ ἀκινησία ἠρεμία ἐστίν, Hence this qualification is added here. This paragraph leads up to the problem discussed in the next chapter: Where does the actualization reside—in the mover or in the moved?—C.]
ARISTOTLE a. A ἢ a 2020 κινητοῦ, ἡ κινητόν: συμβαίνει δὲ τοῦτο θίξει τοῦ κινητικοῦ, ὥσθ᾽ ἅμα Kat πάσχει. ~ Uy i ft Ὁ Εἶδος δὲ del οἴσεταί τι τὸ κινοῦν---ἤτου τόδε ἢ τοιόνδε ἢ τοσόνδε---ὃ ἔσται ἀρχὴ καὶ αἴτιον τῆς ν᾿ t > ij A κινήσεως, ὅταν κινῇ" οἷον 6 ἐντελεχείᾳ ἄνθρωπος ποιεῖ ex τοῦ δυνάμει ὄντος ἀνθρώπου ἄνθρωπον.
CHAPTER ITI ARGUMENT [It follows from the definition of motion that the movement, which is the actualization of both potentialities—that of the mover and that of the moved—occurs in the moved. It is one actualization, with two conceptually distinct aspects Against this there is a specious objection: that there must be two distinct actualizations, active and passive, and if the active is in the agent, the mover will itself be moved; uf both are in the patient, we shall have two different changes in the same thing producing a single result (a 21-86). Whereas if there is only one actualization, how can two conceptually different things have the same actualization? And will not to learn and to teach be the same thing?
Kat τὸ ἀπορούμενον δὲ φανερόν, ὅτι ἔστιν ἡ κίνησις ἐν τῷ κινητῷ" ἐντελέχεια γάρ ἐστι τούτου, 16 καὶ ὑπὸ τοῦ κινητικοῦ. καὶ ἡ τοῦ κινητικοῦ δὲ PHYSICS, III. r.-111.
But since, as we have said, the motion of one body is initiated and sustained by the impact and pressure of another body, and is therefore inci- dentally passive to that body, it is natural that the moving cause of the motion should carry ovex to the thing moved certain characteristies of its own that fall under such categories as substance or quality or quantity, which become principles and determinants of the character of the change (within the potentialities of the subject) which it produces by its causal action. Thus the man in actuality produces the actual man out of something which has the potentiality of becoming man.
CHAPTER III ARGUMENT (continued) These objections are shown not to be substantial (Ὁ 5-22).
The formula: by which we have defined movement can easily be applied to each special sort of movement. The treatment of movement is concluded with a final definition Tue question where the actual movement is realized is answered easily, for it obviously consists in that which was in potentiality capable of motion being now in the realized ‘ act of motion’; and this realized ‘ act of motion’ is caused by what was in potentiality ‘capable of causing’ it now realizing that potentiality by actually moving it. So that the one ‘ act of moving’ in the movable is the realization, ARISTOTLE ~ pS cd 2028 ἐνέργεια οὐκ ἄλλη ἐστίν. δεῖ μὲν γὰρ εἶναι ἐντελέ- χειαν ἀμφοῖν" κινήτικον μεν yap EOTL TG) OVVACUMLL, a ‘ atcy ὡς 3} ἡ 3 \ Αι Κινρὺν δὲ τῳ EVEPyELV, ἀλλ ἐστι» ενεργήτικον TOU κινητοῦ, ὥστε ὁμοίως μία ἡ ἀμφοῖν ἐνέργεια ὠσπέρ TO GaUTO διάστημα ἐν προς vo Kat vo 20 πρὸς εν, καὶ TO ἄναντες Καὶ τὸ KATAVTES’ ταῦτα γὰρ ἕν μὲν ἔστιν, 6 μέντοι λόγος οὐχ εἷς. ὁμοίως δὲ καὶ ἐπὶ τοῦ κινοῦντος καὶ κινουμένου. Ἔχει δ᾽ ἀπορίαν λογικήν: ἀναγκαῖον γὰρ ἴσως εἶναί twa ἐνέργειαν ἄλλην τοῦ ποιητικοῦ καὶ τοῦ παθητικοῦ" τὸ μὲν δὴ ποίησις, τὸ δὲ πάθησις, ἔργον δὲ καὶ τέλος τοῦ μὲν ποίημα, τοῦ δὲ πάθος. 25 ἔπε οὖν ἄμφω Κινήσεις, εἰ μὲν ETEPAL, ἐν TLL; 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). ἀλλὰ μὴν εἰ τοῦτο, ἡ κίνησις 4 SLiterally, ‘ for it must be the realization of both.’—C.]
PHYSICS, III. m1.
not only of its own capacity of being moved, but of the capacity of the potential mover lo be actually moving it. For potential mover and_ potential moved must cach have its proper realization or ‘being in act,’ ὁ for to be a potential mover is to be capable of moving something, but to be an actual mover is to be in the ‘ act of moving’ the something. Now this act is accomplished in the movement of the thing moved, so that the actualizing of the two potentialities coincides (objectively and materially) in the one movement; just as the ratio 1-2 is the same as the ratio 2:1, and the slope of a line is the same whether you regard it as sloping up or down: for in these cases we are dealing with one and the same thing which may be regarded or defined from two different approaches. So too with the mover and the moved.® A specious difficulty may, however, be urged. It seems hard to deny that the energizing of a potential activity must be different from that of a potential passivity; so the one will be a doing and the other a suffering (‘ passion ’), and the actuality and goal of the one is a deed and of the other an experience. Since, then, both are movements, (a) if they are two movements and not one, where are they? Obviously one of them is in the moved. Either then (1) they are both in the moved, or ‘ patient,’ or (2) the one that is a doimg is in the agent and the one that 1s a suffering in the patient (and it will not do to say that this too is a doing, for that would be using words equivocally). In this case (2) the ‘movement’ (which we must regard as a ‘ suffering,’ wherever it 1s seated, or it would not be a movement 5 See Gen. Introd. p. xaviil, on magnet and needle. VOL. I Ρ 200 ARISTOTLE 808ι ἐν τῷ κινοῦντι ἔσται: ὁ γὰρ αὐτὸς λόγος ἐπὶ 80 κινοῦντος καὶ κινουμένου, ὥστ᾽ ἢ πᾶν τὸ κιοῦν κινήσεται, ἢ ἔχον κίνησιν οὐ κινήσεται. εἰ ἄμφω ἐν τῷ κινουμένῳ καὶ πάσχοντι--καὶ ἡ ποίησις καὶ ἡ πάθησις---καὶ ἡ δίδαξις καὶ ἡ μάθη- σις δύο οὖσαι ἐν τῷ μανθάνοντι, πρῶτον μὲν ἡ ἐνέργεια ἡ ἑκάστου οὐκ ἐν ἑκάστῳ ὑπάρξει, εἶτα 85 ἄτοπον τὸ δύο κινήσεις ἅμα κινεῖσθαι" τίνες yap ἔσονται ἀλλοιώσεις δύο τοῦ ἑνὸς καὶ εἰς ἕν εἶδος; ἀλλ’ ἀδύνατον, ἀλλὰ μία ἔσται ἡ ἐνέρ- goab yea. ἀλλ᾽ ἄλογον δύο ἑτέρων τῷ εἴδει τὴν αὐτὴν καὶ μίαν εἶναι ἐνέργειαν: καὶ ἔσται, εἴπερ ἡ δίδαξις Kal ἡ μάθησις ταὐτὸ Kal ἡ ποίησις καὶ ε J, an ἡ πάθησις, καὶ τὸ διδάσκειν τῷ μανθάνειν ταὐτὸ Kal τὸ ποιεῖν τῷ πάσχειν, ὥστε τὸν διδάσκοντα ὃ ἀνάγκη ἔσται πάντα μανθάνειν καὶ τὸν ποιοῦντα πάσχειν.
δ Ἢ οὔτε τὸ τὴν ἄλλου ἐνέργειαν ἐν ἑτέρῳ εἶναι 4 See Book VIII. ch. v.
> [Literally, ‘ how can there be two particular qualitative changes of one thing terminating in a single form?’ referring to the illustration μάθησις, Cf, De anim. 417 a 81 ὁ διὰ μαθήσεως ἀλλοιωθείς, though (as Philoponus there remarks) μάθησις might better be regarded as a γένεσις than as an ἀλλοίωσις,---Ο,] PHYSICS, IIT, ut, at all) in which the agent realizes its potentiality must reside in the mover. Thus every mover must, as such, be in motion *—unless we are to say that a thing may have motion and yet be motionless; which is not the case. If, on the other hand, (1) the two movements—that which actualizes the poten- tiality of the agent and that which actuahzes the potentiality of the patient—both take place in the moved or patient; if, for instance, the process which realizes the power of teaching and that which realizes the power of learning, being two distinct processes, both take place in the learner, then, m the first place, this would be to say that the two ‘potentials’ are not realizing their several po- tentialilies each in itself; and, in the next place, it would involve the paradox of the same thing ex- periencing two movements or progressions at the same time—which is only possible if the progressions differ generically, e g. one being from place to place and the other from quality to quality.’ That is impossible: the actualization must be one. But (ὁ) if it is one, then it may be urged that it 1s mon- strous to assert that two capacities that are conceptu- ally different can find one and the same realization; and if the actualizing of teaching and the actualiz- ing of learning, or that of doing and that of suffering, coincide, then to be teaching and to be learning will be the same thing, and to be acting will be the same as being subject to another’s action. At that rate every one who was teaching would have to be learning all the things he taught, and every one who was doing anything would be having the same thing done to him all the while.
But this is only a quibble. For there is nothing ARISTOTLE 202b ἄτοπον (ἔστι γὰρ ἡ δίδαξις ἐνέργεια τοῦ διδασκαλικοῦ, ἐν τινὶ μέντοι, καὶ οὐκ ἀποτετμημένη, ἀλλὰ τοῦδε ἐν τῷδε)" οὔτε μίαν δυοῖν τὴν αὐτὴν εἶναι κωλύει, μὴ ὡς τὸ εἶναι τὸ αὐτὸ ἀλλ᾽ ὡς 10 ὑπάρχει TO δυνάμει ov πρὸς τὸ ἐνεργοῦν, οὔτ ἀνάγκη τὸν διδάσκοντα μανθάνειν, οὐδ᾽ εἰ τὸ ποιεῖν Kal πάσχειν τὸ αὐτό ἐστι, μὴ μέντοι ὡς τὸν λόγον εἶναι ἕνα τὸν TO τί ἦν εἶναι λέγοντα (ὡς λώπιον καὶ ἱμάτιον) ἀλλ᾽ ὡς ἡ ὁδὸς ἡ Θήβηθεν ᾿Αθήναζε καὶ ἡ ᾿Αθήνηθεν εἰς Θήβας, ὥσπερ 16 cipyrat καὶ πρότερον" οὐ γὰρ ταῦτα πάντα ὑπάρχει rots ὁπωσοῦν τοῖς αὐτοῖς, ἀλλὰ μόνον ols τὸ εἶναι τὸ αὐτό. οὐ μὴν ἀλλ᾽ οὐδ᾽ εἰ ἡ δίδαξις τῇ 2 μαθήσει τὸ αὐτό, καὶ τὸ μανθάνειν τῷ διδάσκειν" ὥσπερ οὐδ᾽ εἰ ἡ διάστασις μία τῶν διεστηκότων, καὶ τὸ διίστασθαι ἐνθένθε ἐκεῖσε κἀκεῖθεν δεῦρο a ἐν καὶ τὸ αὐτό. ὅλως δ᾽ εἰπεῖν οὐδ᾽ ἡ δίδαξις τῇ μαθήσει οὐδ᾽ ἡ ποίησις τῇ παθήσει τὸ αὐτὸ κυρίως, ἀλλ᾽ ᾧ ὑπάρχει ταῦτα-- κίνησις' τὸ γὰρ PHYSICS, II. mr.
paradoxical in the actualizing of the potentiality of one thing having its effective seat in another (for the potentiality of an agent—the ‘ teacher ’ for instance —is actualized not σὲ vacuo or in isolation, but in the ‘taught’: it is the actualizing of this zz that); nor is there anything to prevent the same one thing being the actnalizing of two potentialities—not in the sense of unqualified identity, but in the sense of being in the one relation of actuality to potentiality with regard to the two potentialities So it does not follow that the teacher must be learning, even though we admit that action and passion coincide, not in the sense of being the same in the definitions that determine them (like ‘ garments ’ and‘ clothes ἢ, but in the sense in which (to vary the illustrations already given “) the road from Thebes Lo Athens is the same as the road from Athens to Thebes; for things need not be identical in all respects because they are the same in some, but only if they are identical in what they actually are—in a word if they are not two ‘things ’ at all, but only two names or definitions of the same thing. Nor yet, even if the actualizing of teaching is identical with the actualizing of learning, does it follow that to be learning is the same thing as to be teaching; any more than, if the interval between two distant points 4 and B is one interval, it follows that being distant from B when you are at 4 is the same as being distant from d when you are at B. To put it generally; there is, in the strict sense, no absolute identity even between the actualization of teaching and that of learning, or between action and passion; but only the thing which has these two aspects—the motion—is one and the same thing; for there is a conceptual ARISTOTLE 205} τοῦδε ἐν τῷδε καὶ τὸ τοῦδε ὑπὸ τοῦδε ἐνέργειαν εἶναι ἕτερον τῷ λόγῳ. Ti μὲν οὖν ἐστι κίνησις εἴρηται καὶ καθόλου καὶ κατὰ μέρος--οὐ γὰρ ἄδηλον ὡς ὁρισθήσεται 5 τῶν εἰδῶν ἕκαστον αὐτῆς- ἀλλοίωσις μὲν γὰρ ἡ τοῦ ἀλλοιωτοῦ, ἧ ἀλλοιωτόν, ἐντελέχεια---ἔτι δὲ γνωριμοωύτερον, ἡ τοῦ δυνάμει ποιητικοῦ καὶ παθητικοῦ, ἢ τοιοῦτον, ἁπλῶς τε καὶ πάλιν καθ᾽ ἕκαστον, ἢ" οἰκοδόμησις ἣ ἰάτρευσις. τὸν αὐτὸν δὲ λεχθήσεται τρόπον καὶ περὶ τῶν ἄλλων κινήσεων ἑκάστης.
CHAPTER IV ARGUMENT [The study of Infinity, as involved in space, motion, and time, belongs to Physics (202 b 30-86).
Earlier philosophers all discuss the Unlimited. (1) The Pythagoreans and Plato treat ‘ the Infinite itself’ as having an independent substantive ewistence(b 36-203 0 16). (2) The physicists believe in an infinite element (the Ionian monists), or an infinite mass of particles. These, according to Anan- agoras, originally formed an infinite miature before differen- tiation began. Democritus’ atoms are eternally distinct, but the ‘ body’ of which they are all composed is a sort of material £0 πεὶ ἐστὶν ἡ περὶ φύσεως ἐπιστήμη περὶ a Ν μεγέθη καὶ κίνησιν καὶ χρόνον, ὧν ἕκαστον a a ἀναγκαῖον ἢ ἄπειρον ἢ πεπερασμένον εἶναι--εἰ 1 [For ἢ E has ὅτι (according to Bekker).—C,] PHYSICS, II. u1-1v.
distinction between the aspecls—between ‘ being the actualizing of this in that’ and ‘being the actualizing of that by this.’
We have now explained the nature both of move- ment in general and of the special kinds of movement (for it is easy to see how each special kind is to be defined; qualitative alteration, for example, as the actualization of what has the potentiality of this kind of alteration, in so far as it has that potentiality); or to put it still more clearly: motion is the actualiza- tion of the potentially active and of the potentially passive, as such—as a general formula and again as applied to any particular case, the process of building or of healing. Each of the other kinds of motion will be described in similar terms.
CHAPTER IV ARGUMENT (continued) principle, indeterminate when considered in abstraction from atomic forms (a 16-b 2).
All agree that the Infinite is a‘ beginning’ or principle, which the earliest physicists regarded as a divine, self- moving, element (b 3-16).
Five considerations which lead to a belief in something that is infinite (Ὁ 15-80).
Tae probleme that arise about its nature (Ὁ 30-204 a 2).
The senses in which ‘infinite’ is used (a 2-7 =Met.
Tue study of Nature is concerned with extension, motion, and time; and since each one of these three must be either limited or unlimited (which is not to 91 ARISTOTLE soap Kal μὴ πᾶν ἐστιν ἄπειρον ἢ πεπερασμένον, οἷον πάθος ἢ στιγμή; τῶν γὰρ τοιούτων ἴσως οὐδὲν ἀναγκαῖον ἐν θατέρῳ τούτων εἶναι--προσῆκον ἂν a εἴη τὸν περὶ φύσεως πραγματευόμενον θεωρῆσαι περὶ ἀπείρου, εἰ ἔστιν ἢ μή, καὶ εἰ ἔστι, τί ἐστιν. 2032 Σημεῖον δ᾽ ὅτι τῆς ἐπιστήμης οἰκεία ἡ περὶ αὐτὸ θεωρία' πάντες γὰρ οἱ δοκοῦντες ἀξιολόγως ἦφθαι τῆς τοιαύτης φιλοσοφίας πεποίηνται λόγον περὶ τοῦ ἀπείρου, καὶ πάντες ὡς ἀρχήν τινα τιθέασι τῶν ὄντων,-- Οἱ μέν, ὥσπερ οἱ Πυθαγόρειοι καὶ Πλάτων, καθ᾽ ε αὑτό, οὐχ ὡς συμβεβηκός τινι ἑτέρῳ ἀλλ᾽ οὐσίαν αὐτὸ ὃν τὸ ἄπειρον. πλὴν οἱ μὲν Πυθαγόρειοι ἐν τοῖς αἰσθητοῖς (οὐ γὰρ χωριστὸν ποιοῦσι τὸν ἀριθμόν), καὶ εἶναι τὸ ἔξω τοῦ οὐρανοῦ τὸ ἄπειρον. Πλάτων δὲ ἔξω μὲν οὐδὲν εἶναι σῶμα, οὐδὲ τὰς ἰδέας, διὰ τὸ μηδέ που εἶναι αὐτάς, τὸ μέντοι ἄπειρον καὶ τὸ ἐν τοῖς αἰσθητοῖς καὶ ἐν ἐκείναις εἶναι. καὶ οἱ μὲν τὸ ἄπειρον εἶναι τὸ ἄρτιον' τοῦτο γὰρ ἐν- απολαμβανόμενον καὶ ὑπὸ τοῦ περιττοῦ περαινό- μενον παρέχειν τοῖς οὖσι τὴν ἀπειρίαν: σημεῖον δ᾽ εἶναν τούτου τὸ συμβαῖνον ἐπὶ τῶν ἀριθμῶν" περιτιθεμένων γὰρ τῶν γνωμόνων περὶ τὸ ἕν καὶ 4 (Or, ‘contained in sensible things’ as a constituent of them; for ‘the elements of Number (Even = Unlimited, Odd =Limit or Limited) are the elements of all things,’ PHYSICS, ΠῚ, τν.
say that everything must be limited or unlimited, for it docs not preclude us from denying the necessity of such things as a‘ point ’ or an‘ experience’ bemg either one or the other), it follows that the student of Nature must consider the question of the unlimited, with a view to determining whether it exists at all, and, if so, what 1s ils nature.
That this mquiry is really germane to our subject is indicated by the fact that all philosophers of repute who have dealt’ with Physics have discussed the ‘ unlimited,’ and all have regarded it as in some sense a‘ principle ’ of actually existing things.
Some, such as the Pythagoreans and Plato, have regarded the unlimited or undetermined as existing in itself, and not as being a condition incident to something else, but having its own independent substantive existence. But the Pythagoreans regard it as something cognizable by the senses ὅ (since they do not regard ‘number’ as having a detached existence apart from the objects of the senses), and held that what lies beyond the heaven is ‘the un- limited’; whereas Plato holds that no material body lies outside the heaven at all—nor any ‘ Ideas’ either, for they cannot be said so much as to be ‘located ’ anywhere; but, for all that, he finds the ‘undetermined’ both in the objects of sense and in the ‘Ideas.’ Again, the Pythagoreans identify the ‘undetermined’ with the ‘even.’ For evenness, when it has been enclosed and determined by the odd unit, is still the principle of the undetermined aspect of things. And this they find reflected in the properties of numbers; for successive additions to the unit of the odd numbers (even numbers gno- monized by the extra unit) preserve the definite quad- ARISTOTLE 8084 10 χωρὶς ὁτὲ μὲν ἄλλο ἀεὶ γίγνεσθαι τὸ εἶδος, ὁτὲ « [The Pythagoreans represented numbers by patterns or figures (εἴδη) of pebbles or counters, called ‘terms’ (ὅροι), as being like terminal stones marking ont a field or space, Which without them would be ‘ unlimited’ and ‘ void.’ Each pebble stands for an indivisible unit (monad). The unit 1s not a number. Numbers are aggregates of two or more monads; so 2 is the first even number, 3 the first odd, (1) τὸ ἄπειρον εἶναι τὸ ἄρτιον. Cf. Met, 986 a 1 and 17, the elements of number (and hence of all things) are the Even, which is Unlimited, and the Odd, which is Limited. According to the authorities quoted by Simphic. (455. 15), the Even is unlimited because the Even is divisible into two equal parts ad infinitem, but the addition of the odd sets a linit to this division. This and similar statements are explained graphically by Heidel (Arch. 7. Gesch, Phil, xiv, 890 ff.). Take any even number, say 10 (figured thus +»), The process of halving, represented by the arrow, goes on indefinitely; but it would be arrested at any pomt by the addition of an odd unit, eg. in 11 (————->-). (Gf. Ross on Met, 986 a 18 and references there.)
there.)
(2) 11 τοῦτο γὰρ...12 ἀπειρίαν, In this clause τοῦτο may mean ‘this principle, the Even= Unlimited,’ and τὸ περιττόν its opposite, ‘the Odd=Limit.’ As applied to numbers, the Unlimited is the ao space enclosed between two Limits (units) set in a row. But, by some Pythagoreans at least, the process of ‘ generating ’’ numbers was not distinguished from the generation of ‘ things ’"—cosmogony,—for ‘ things ἢ (bodies existing in actual space) are ‘ numbers’ (aggregates of monads or atomic bodies). So this clause may be com- pared with the Pythagorean ee process as de- scribed at Met. 1091 a 14, ‘ when the One had been con- strucled,,. the nearest part of the Unlimited began to be constrained and limited by the Limit.’ Here the first monad, wot sine outwards from the centre of an unformed mass, probably, of ‘ air’ (or ‘ void’), introduces shape or limit, and forms a world. This void would (in the terms of our PHYSICS, ΠΙ. 1.
rate form intaet, whereas, if the determining unit is withheld, the succession of even numbers yields a perpetually varying and undetermined figure.* passage) ‘ when enclosed by the Odd (Limit) contribute to things the element of unlimitedness ’ (the vacancy contained by the planes bounding a body).
(3) περιτιθεμένων...ὁτὲ δὲ ἕν. The gnomon is the car- » for measuring a right angle. na When the wneven numbers (3, 5, 7, ete.) are ‘ put round’ the unit in figures of this shape, the result is always the τα] (4) καὶ χωρίς...ἄλλο ἀεὶ γίγνεσθαι εἶδος is explained as meaning that, if even numbers are put in Hi round the first even, 2, the resulting figures are oblongs, all dispenter’s tool, shaped same figure, viz., ἃ square, thus: Arithmetically similar in form—an infinity of forms problem is, how to get this meaning out of καὶ χωρίς. Heath (Gk. Maths. i. 83) and others take χωρίς to mean ‘ in the separate (er other) case.’ Taylor’s explanation (C.R. xl. 150) is unsatisfactory. Burnet (Ε΄. G. Ph! 103) under- stands ‘ χωρὶς τοῦ ἑνός, i.e. starting from 2, not from 1.’ But why say ‘ without the unit’ instead of rept τὰ duo? Possibly ὁ putting gnomons round without the unit’ means putting ¢wo minimum gnomons round one unother, thus Re. Each succeeding gnomon will then contain an even number (6, 8, 10, etc.), and we shall obtain the traditional series of varying oblongs.—C.]
ARISTOTLE 20sa δὲ ἕν. Πλάτων δὲ δύο τὰ ἄπειρα, τὸ μέγα καὶ τὸ μικρόν.
Oi δὲ περὶ φύσεως ἅπαντες' ὑποτιθέασιν ἑτέραν τινὰ φύσιν τῷ ἀπείρῳ τῶν λεγομένων στοιχείων, οἷον ὕδωρ ἢ ἀέρα ἢ τὸ μεταξὺ τούτων. τῶν ᾿δὲ πεπερασμένα ποιούντων στοιχεῖα οὐθεὶς ἄπειρα 90 ποιεῖ" ὅσοι δ᾽ ἄπειρα ποιοῦσι τὰ στοιχεῖα, καθάπερ ᾿Αναξαγόρας καὶ Δημόκριτος, 6 μὲν ἐκ τῶν ὁμοιομερῶν, ὁ δ᾽ ἐκ τῆς πανσπερμίας τῶν σχημά- των, τῇ ἁφῇ συνεχὲς τὸ ἄπειρον εἶναί φασιν.
Καὶ ὁ μὲν ὁτιοῦν τῶν μορίων εἶναι μῖγμα ὁμοίως τῷ παντί, διὰ τὸ ὁρᾶν ὁτιοῦν ἐξ ὁτουοῦν 8 γιγνόμενον. ἐντεῦθεν γὰρ ἔοικε καὶ ὁμοῦ ποτε πάντα χρήματα φάναι εἶναι" οἷον ἥδε ἡ σὰρξ καὶ τόδε τὸ ὀστοῦν, καὶ οὕτως ὁτιοῦν" καὶ πάντα ἄρα" καὶ ἅμα τοίνυν: ἀρχὴ γὰρ οὐ μόνον ἐν ἑκάστῳ ἐστὶ τῆς διακρίσεως, ἀλλὰ καὶ πάντων. ἐπεὶ γὰρ τὸ γιγνόμενον ἐκ τοῦ τοιούτου γίνεται σώματος, a0 πάντων δ᾽ ἐστὶ γένεσις (πλὴν οὐχ ἅμα), καί τινα ἀρχὴν δεῖ εἶναι τῆς γενέσεως. αὕτη δ᾽ ἐστὶ μία, 1 [ἅπαντες] ἀεὶ πάντες EK, ἅπαντες ἀεὶ al. Simple. 468, 17 (lemma) and 459, 8 omits def.—C.]
> [ie whereas Plato and the Pythagoreans talked simply of ‘the Unlimited’ in the abstract, the physicists have an unlimited something. (1) The Ioman monists make this something water, air, etc, Of the pluralists, (2) Empedo- eles has no unlimited, but a finite number of elements, each finite in quantity; (8) Anaxagoras and Democritus have an infinite number of elementary particles, and ‘the Infinite’ means the infinite aggregate of these particles in contact with one another.—C.]
PHYSICS, III. 1.
Plato, for his part, recognizes two aspects of the ‘ unlimited,’ the great and the small.¢ The physicists, on the other hand, all make some other nature—one of their so-called elements, water or air or the intermediate between these—a subject of which ‘ unlimited’ is predicate;® and only those of them, such as Anaxagoras and Democritus, who hold the elements themselves to be unlimited, admit the principle of the unlimited at all. The others, who regard the ‘elements’ as limited, will not allow that anything at all is unlimited; whereas Anaxagoras finds the unlimited in his ‘ parts-after- their-kind,’ and Democritus in his hotch-potch of multi-shaped atoms; while both of them regarded it as a mass having continuity by contact of unlike particles.
Anaxagoras, moreover, held that any part whatso- ever was as much a mixture as the whole, because he saw that any one thing comes into being out of any other thing. For this seems to be the original ground of his assertion that ‘ all things are together ’ at some moment. ‘Take, for instance, this piece of flesh and this piece of bone: either can come out of the other; and so with any one thing; and accordingly, αἰῤ things must come out of ald things, and therefore all things must be together at the same moment; for there must be a beginning of the separating out, not only in each thing, but of all things in general. For since the thing that comes into separate existence comes out of that in which it exists unseparated, and everything thus comes out of the unseparated (though successively, not all at once), there must be some initiating principle of ‘coming out of the undistinguished’ in general.
ARISTOTLE 208a dy ἐκεῖνος καλεῖ νοῦν' ὁ δὲ νοῦς dm’ ἀρχῆς τινος ἐργάζεται νοήσας, ὥστε ἀνάγκη ὁμοῦ ποτε πάντα εἶναι καὶ ἄρξασθαί ποτε κινούμενα. Δημόκριτος δ᾽ οὐδὲν ἕτερον ἐξ ἑτέρου γίγνεσθαι τῶν πρώτων 808 φησίν: ἀλλ᾽ ὅμως γε αὐτῶν τὸ κοινὸν σῶμα πάντων ἐστὶν ἀρχή, μεγέθει κατὰ μόρια καὶ σχή- ματι διαφέρον.
Ὅτι μὲν οὖν προσήκουσα τοῖς φυσικοῖς ἡ θεωρία, δῆλον ἐκ τούτων. εὐλόγως δὲ καὶ ἀρχὴν υ αὐτὸ τιθέασι πάντες" οὔτε γὰρ μάτην αὐτὸ οἷόν τε εἶναι, οὔτε ἄλλην ὑπάρχειν αὐτῷ δύναμιν πλὴν ὡς ἀρχήν! ἅπαντα yap ἢ ἀρχὴ ἢ ἐξ ἀρχῆς, τοῦ δὲ ἀπείρου οὐκ ἔστιν ἀρχή" εἴη γὰρ dv αὐτοῦ πέρας. ἔτι δὲ καὶ ἀγένητον καὶ ἄφθαρτον ὡς ἀρχή τις οὖσα' τό τε γὰρ γενόμενον ἀνάγκη τέλος 1ὸ λαβεῖν, καὶ τελευτὴ πάσης ἐστὲ φθορᾶς. διὸ (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 α [In this paragraph the meaning of ἀρχή shifts, The ‘unlimited ' of Anaxagoras considered as the initial state of matter (ἀρχή) is an unhmited confused mass, actuall existing as such before the moment of time when Min makes a beginning (ἀρχὴ) of the ‘ separating-out.’ This 1s contrasted with Democritus’s multitude of immutubly distinct atoms; but Aristotle suggests that ‘body’, of which these atoms all consist, may be regarded as a sort of indeterminate and ‘unlimited ’ material principle (ἀρχὴ), though it never actually exists by itself. So both Anaxagoras and Democritus recognize an ‘unlimited’ (or ‘ indeter- minate ἢ principle (d4px4).—C.]
ν [The archaic form of this traditional argument recalls Plato's proof of the immortality of the self-moving soul: Phaedrus 245 p ἀρχὴ δὲ ἀγένητον" ἐξ ἀρχῆς yap ἀνάγκη πᾶν τὸ PHYSICS, III. w.
And this principle is one, and is what he designates as ‘Intelligence’; which Intelligence must have made some beginning of its action as such; so there must have been a time when all things were indis- tinguishably one, and a point of time at which the stirring of segregative movement began. Demo- critus, on the other hand, will not have it that any of his ultimate atoms can proceed out of any other; but nevertheless the matter itself that is distinguished in the different kinds of atom by size and shape, is a common principle that underlies them all.4 Clearly then the study of the undetermined or unlimited is germane to that of Nature. And again, all those who accept it are quite right in regarding it as a ‘principle’; for, if it exists, it must affect things somehow, and it cannot affect them except as a principle; for everything is either determined by some principle or is a principle itself, and the undetermined cannot be determined at all, and so cannot depend upon anything else as its principle. And further, being a principle, it can have no be- ginning or end of existence; for whatever comes into being must come to an end, and there must be a term to any process of perishing.’ So the ‘unlimited’ cannot be derived from any other principle, bui is ilself regarded as the principle of the other things, ‘embracing and governing all,’ as it is said to do by such as accept it, unless indeed they accept other principles alongside of it, such as ‘ Intelligence’ or ἡ Amity.’ This unlimited, then, γιγνόμενον γίγνεσθαι, αὐτὴν δὲ μήδ᾽ ἐξ ἑνός, εἰ γὰρ ἔκ τον ἀρχὴ γίγνοιτο, οὐκ ἂν ἔτι ἀρχὴ γίγνοιτο. ἐπειδὴ δὲ ἀγένητόν ἐστιν, καὶ ἀδιάφθορον...Rostagni ae di Pitagora, 137), suggests that Plato was echoing Alemaeon, but Anaximander may have argued similarly.-—C.]
ARISTOTLE a ial f 3 203» τοῦτ᾽ εἶναι τὸ θεῖον: ἀθάνατον yap καὶ ἀνώλεθρον, iz a wis φησιν ὃ ᾿Αναξίμανδρος καὶ of πλεῖστοι τῶν φυσιολόγων. Tod δ᾽ εἶναί τι ἄπειρον ἡ πίστις ἐκ πέντε ῃ? ny, κι ” ὡς μάλιστ av συμβαίνοι σκοπουσιν' €K TE TOU χρόνου (οὗτος γὰρ ἄπειρος)" καὶ ἐκ τῆς ἐν τοῖς μεγέθεσι διαιρέσεως (χρῶνται yap καὶ οἱ μαθη- ματικοὶ τῷ ἀπείρῳ)" ἔτι τῷ οὕτως ἂν μόνως μὴ ὑπολείπειν γένεσιν καὶ φθοράν, εἰ ἄπειρον εἴη 20 ὅθεν ἀφαιρεῖται TO γιγνόμενον" ἔτι τῷ TO πεπερα- σμένον ἀεὶ πρός τι περαίνειν, ὥστε ἀνάγκη μηδὲν rE > a εἶναι πέρας, εἰ del περαίνειν ἀνάγκη ἕτερον πρὸς ἕτερον" μάλιστα δὲ καὶ κυριώτατον, ὃ τὴν κοινὴν a \ a ποιεῖ ἀπορίαν πᾶσιν" διὰ γὰρ τὸ ἐν τῇ νοήσει μὴ 26 ὑπολείπειν, Kal ὁ ἀριθμὸς δοκεῖ ἄπειρος εἶναι καὶ τὰ μαθηματικὰ μεγέθη Kat τὸ ἔξω τοῦ οὐρανοῦ. ἀπείρου δ᾽ ὄντος τοῦ ἔξω, καὶ σῶμα ἄπειρον εἶναι δοκεῖ καὶ κόσμοι' τί γὰρ μᾶλλον τοῦ κενοῦ ἐνταῦθα ἢ ἐνταῦθα; ὥστ᾽ εἴπερ μοναχοῦ, Kat πανταχοῦ εἶναι τὸν ὄγκον. ἅμα δ᾽ εἰ καὶ ἔστι κενόν, καὶ τόπος ἄπειρος, Kal σῶμα ἄπειρον εἶναι 2 (Diels, Vors. Anaximandros 15. The φυσικοί who have not distincl moving causes (Anaxagoras’ νοῦς, Empedocles’ φιλία) regard the primary stuff of things as sclf-moving, 1.6, animate with a divine (immortal) life, and ‘governing’ the PHYSICS, IIT. rv.
would be the divinity itself, being ‘immortal and indestructible,’ as Anaximander and most of the physicists declare it to be.* Now the belief in the existence of something ‘unlimited ’ appears to rest, in the main, upon the consideration of five things: (1) Time (which is regarded as having no limit); (2) the divisibility of magnitudes (which is treated by mathematicians as possible without limit); (8) the idea that from the unfailing persistence of genesis and its opposite it follows that the things which come into being are drawn from an unlimited store; or again (4) the argument that, since whatever is limited reaches its limit by coming up to something else, there ean be no absolute limit, for nothing can be limited except by something else beyond its hmit. But the most effective and central consideration, the weight of which all thinkers alike have felt, is that (5) the imagination can always conccive a‘ beyond ’ reaching out from any limit, so that the series of numerals seems to have no limit, nor mathematical magnitudes, nor the ‘ beyond the heavens.’ Moreover, it seems to follow from the ‘ beyond’ being unlimited, that ‘body ’ must be unlimited, and that there must be unlimited worlds. For why should there be more vacancy in one place than in another? So, if mass ὃ is anywhere, must it not be everywhere? Or even if you assume vacancy everywhere, does it not follow that there is ‘ place ’ everywhere, and ‘ body ’ movement of the finite things it ‘ embraces’ (συγκρατεῖ, περιέχει eas frag. 2: ἐκυβέρνησε Heracleitus, frag. ΑἹ, etc.).—C.]
> [ὄγκος was used as synonym of ‘ atom’ by Democritus (Diog. Lacrt. ix. 44). Aristotle refers to the unlimited worlds of the Atomists.—C.]
VOL. I Q 225 ARISTOTLE 203 80 ἀναγκαῖον" ἐνδέχεσθαι γὰρ ἢ εἶναι οὐδὲν διαφέρει ἐν τοῖς ἀϊδίοις.
"Exes δ᾽ ἀπορίαν ἡ περὶ τοῦ ἀπείρου θεωρία" καὶ γὰρ μὴ εἶναι τιθεμένοις πόλλ᾽ ἀδύνατα συμ" βαίνει, καὶ εἶναι’ ἔτι δὲ ποτέρως ἔστι---πότερον ὡς οὐσία, ἣ ὡς συμβεβηκὸς Kal? αὐτὸ φύσει τινί, ἢ οὐδετέρως, ἀλλ᾽ οὐδὲν ἧττόν ἐστιν ἄπειρον ἢ 24a ἄπειρα τῷ πλήθει. μάλιστα δὲ φυσικοῦ ἐπι- σκέψασθαι, εἰ ἔστι μέγεθος αἰσθητὸν ἄπειρον.
Πρῶτον οὖν διοριστέον ποσαχῶς λέγεται τὸ ἄπειρον. ἕνα μὲν δὴ τρόπον τὸ ἀδύνατον δι- ελθεῖν τῷ μὴ πεφυκέναι διιέναι (ὥσπερ ἡ φωνὴ ἀόρατος)" ἄλλως δὲ τὸ διέξοδον ἔχον ἀτελεύτητον, ἢ ὃ μόλις, ἢ ὃ πεφυκὸς ἔχειν μὴ ἔχει διέξοδον ἢ πέρας. "Er ἄπειρον ἅπαν ἢ κατὰ πρόσθεσιν, ἢ κατὰ διαίρεσιν, ἢ ἀμφοτέρως.