> [More literally, ‘ Or does the same thing (above said about ‘ sharp’) hold also of these terms (‘ much’ and ‘double ’)—for “much” too is an ambiguous word—only (with this difference) that in some cases (such as * much’ and ‘double,’ not only the words, as in the case of ‘ sharp,’ but) also the definitions of the words are ambiguous ὃ ’—C.]
PHYSICS, VII. rv.
would cover an equal distance in less time, so that there must be some portion of the time A in which B traverses a part of the arc β, equal to the whole line c, which C takes the whole of the time A to traverse. But if the two can be so equated, we arrive at the same conclusion, namely that an arc may be equal to a straight line. But they cannot be compared, and therefore our first impression, that rates of motion through them can be so compared, is false.
Then (it may be thought) things can be compared if they can be described without equivocation as of the same order. Thus: Why can we not say which is sharpest, this pencil, this wine, or this musical note? Because ‘ sharp’ is equivocal, so the several sharpnesses cannot be compared; but we can compare the highest note in the scale with the leading note, because “ sharp’ means the same thing for both. May it be, then, that the word ‘ quick’ has not the same meaning when applied to rectilinear motion as when applied to circular motion, and still less when one movement is a qualitive modification and the other alocalmovement? Butdoesthis take us far enough ἢ Or are there even some cases in which things that bear the same name without equivocation are, never- theless, not comparable? Take the term ‘ much.’ In ‘much water,’ and ‘much air,’ ‘much’ has the same meaning; but you cannot say whether this much air is equal to this much water, or which is greater. Or if you deny that ‘much’ means the same in both cases, at any rate * double ’ would seem to admit no ambiguity, for in every case it is the ratio of two to one. Yet here twice as much water and twice as much air are not comparable.* ὃ But then too the same argument employed above will show ARISTOTLE ARISTOTLE 8480 λόγος--καὶ γὰρ τὸ πολὺ ὁμώνυμον---ἀλλ᾽ ἐνίων καὶ of λόγοι ὁμώνυμοι, οἷον εἰ λέγοι τις ὅτι τὸ πολὺ TO τοσοῦτον Kal ἔτι, ἄλλο TO τοσοῦτον" Kal TO ἴσον ὁμώνυμον’ καὶ TO ἕν δέ, εἰ ἔτυχεν, εὐθὺς ο ὁμώνυμον, εἰ δὲ τοῦτο, καὶ τὰ δύο. ἐπεὶ διὰ τί sy \ Ἁ \ f ey ’ τὰ μὲν συμβλητὰ τὰ δὲ οὔ, εἴπερ ἦν μία φύσις; ὅτι ἐν ἄλλῳ πρώτῳ δεκτικῷ; ὁ μὲν οὖν ἵππος Kat 6 κύων συμβλητά, πότερον λευκότερον" ἐν @ yap πρώτῳ, TO αὐτό, ἡ ἐπιφάνεια" καὶ κατὰ ο5 μέγεθος ὡσαύτως: ὕδωρ δὲ Kal φωνὴ οὔ' ἐν ἄλλω ᾽ an onA 4 3 Ψ / 4 » yap. ἢ δῆλον ὅτι ἔσται οὕτω γε πάντα ἕν 8496 ποιεῖν, ἐν ἄλλῳ δὲ ἕκαστον φάσκειν εἶναι, καὶ \ ἔσται ταὐτὸν ἴσον καὶ γλυκὺ Kal λευκόν, ἀλλ᾽ ἄλλο α [For τὸ τοσοῦτον καὶ ἔτι in this sense the Oxf. Trans. compares Met. 1021 a 6 τὸ δὲ ὑπερέχον πρὸς τὸ ὑπερεχόμενον τοσοῦτόν τέ ἐστι καὶ ἔτι, Cf. also De caelo 273 Ὁ 81.---Ο,] > [The ambiguity of terms expressing differences of quan~ tity has suggested that the cause of incomparabulity lies in the natures or constitutions (as distinct from the magnitudes or volumes) of the substances concerned. This suggestion is pursued in the next paragraph, but will be found inadequate.
¢ {Simplicius takes the objection to be that, if we make the cause of incomparability reside always and solely in a differ- ence of kind between the subjects having the attribute, then we shall do away with the distinction between ambiguous and unambiguous attributes used above (248 b 7) to explain some cases of incomparability: we shall no longer say that ‘ sweet’ has different meanings as applied to water and to a voice. Translate: ‘ But obviously on this showing it will be possible to make out that all attributes are unambiguous (év) and to allege (as a reason for incomparability, merely) that any one PHYSICS, VII. rv.
PHYSICS, VII. rv.
that if ‘much’ cannot bring water and air on to a scale of comparison, neither can double. The fact is that sometimes definitions themselves are am- biguous. Thus if ‘much’ be defined as ‘so much and more,’ * ‘so much’ will mean a different amount in different cases; and ‘equal’ or ‘the same amount’ is similarly ambiguous, and (‘equal’ being the ratio of one to one) that at once implies that it may very well happen that ‘one’ is ambiguous, and if ‘one’ is ambiguous, so is ‘two’ or ‘double.’ For why, when some things are comparable, should others not be so, if there were no difference in the nature of the things of which the various amounts are taken? ὃ Shall we say it is because the direct and primary seat of the attributes in question is different in kind? Thus we can say whether a horse or a dog is the whiter, for the seat of the colour is the same in both cases, to wit the surface; and in like manner you can say which is greater, because magnitude or volume in each case is the measure of body; but you cannot say whether this volume of water is greater or less than this volume of sound, for one volume is material and the other qualitive. ¢°But obviously if you take difference in the recipient to be the sole cause of incomparability, you will be driven to the doctrine of ideal unities and material diversities, so that there will be an absolute ‘same’ and ‘ equal’ and ‘sweet’ and ‘white,’ of which diverse recipients partake of them is in a subject that is different in kind: ‘ equal’ or ‘sweet’ or ‘ white’ will be unambiguous, only different in- stances of any one of them will be in subjects different in kind. (This is absurd;) and moreover, not any and every subject is capable of a given attribute, but one attribute belongs to one kind of subject in the primary sense.’ (Water is not capable of the kind of sweetness that can belong to a voice.)—C. ] ARISTOTLE 5494 ἐν ἄλλῳ. ἔτι δεκτικὸν οὐ TO τυχόν ἐστιν, ἀλλ᾽ ἕν ἑνὸς τὸ πρῶτον. ᾿Αλλ᾽ ἄρα οὐ μόνον δεῖ τὰ συμβλητὰ μὴ ὁμώνυμα εἶναι ἀλλὰ καὶ μὴ ἔχειν διαφορὰν μήτε ὃ μήτ᾽ ὅ ἐν ᾧ; λέγω δὲ οἷον χρῶμα ἔχει διαίρεσιν: τοιγαροῦν οὐ συμβλητὸν κατὰ τοῦτο, οἷον πότερον κεχρωμά- τισται μᾶλλον (μὴ κατά τι χρῶμα ἀλλ᾽ ἧ χρῶμα), ἀλλὰ κατὰ τὸ λευκόν. οὕτω καὶ περὶ κίνησιν ὁμοταχὲς τὸ ἐν ἴσῳ χρόνῳ κινηθὲν ἴσον τοσονδί. 10 εἰ δὴ τοῦ μήκους ἐν τῳδὶ τὸ μὲν ἠλλοιώθη τὸ δὲ ἠνέχθη, ἴση ἄρα αὕτη ἡ ἀλλοίωσις καὶ ὁμοταχὴς τῇ φορᾷ; ἀλλ᾽ ἄτοπον" αἴτιον δ᾽ ὅτι ἡ κίνησις ἔχει εἴδη. ὥστε εἰ τὰ ἐν τῷ ἴσῳ χρόνῳ ἐνεχθέντα ἴσον μῆκος ἰσοταχῆ ἔσται, ἴση ἡ εὐθεῖα καὶ ἡ περιφερής. πότερον οὖν αἴτιον, ὅτι ἡ φορὰ γένος 7 ὅτι ἡ 15 γραμμὴ γένος; (6 μὲν γὰρ χρόνος ὁ αὐτός. ἂν α See Vol. I. Introd. Ὁ. xliv.
δ And the impossibility of this consequence proves that the antecedent supposition of their equal velocities is also impossible. [The Greek is very obscure, and the Oxf. Trans. suggests alacuna. Itis certainly hard to get out of ὥστε εἰ, κτλ, the required meaning, which seems to be as follows: ‘ And so (if we eliminate such a difference in kind between the move- ments and take a case where both movements are of the same generic kind—locomotion, and) if we are to say that things which move locally over an equal distance in the same time are of equal velocity, then (on the supposition that straight- moving and circular-moving bodies can be of equal velocity, we get the impossible conclusion:) the rectilinear distance will be equal to the circular distance. Which, then, of two possible ambiguities is responsible for this impossible con- clusion—is it the fact that ““ locomotion ” is a genus (not an indivisible species), or is it the fact that “line” (track) is a genus? (It must be one or the other or both,) for time is PHYSICS, VII. rv.
diversely.¢ Moreover, we must note that a receptive factor of one kind cannot receive any impress at random, but each such receptive factor is susceptible only of one corresponding order of participation. May we then conclude that, in order to be com- parable with each other, the terms compared must not only be unequivocal, but must also have no specific differentia from each other either in them- selves or in that in which they are manifested? I mean that ‘colour’ for instance, is susceptible of specific division. Therefore you cannot say which of two things is the more ‘ coloured,’ in the generic sense of ‘colour’; but you can say which is the whiter. So too in the case of motion or passing: two things are moving at the same rate if they take an equal time to accomplish a certain equal amount of motion. Suppose then that one half of a body’s length suffers a qualitive modification and the other a change of place in a certain time, is the modification of the one equal to the local move- ment of the other, and their velocities equal? Mani- festly not; and the reason is that there are distin- guishable kinds of passing from this to that. So that if it were really true that a mobile moving on a circular track and one moving on a rectilinear one could have an equal velocity, then a circular and a rectilinear track might be equal.? Why then cannot they have equal velocity? Is it that ‘local movement’ is a genus and not a species, or that ‘line’ (track) is?
(It must be one or the other or both, for ‘time’ is certainly an unequivocalterm.) Both the movement certainly not divisible into species. (It is in fact both; for) if there are specific differences of “line” (track), there must be specific differences of locomotion to correspond.’ Cf.
20 25 ARISTOTLE δὲ τῷ εἴδει ἢ ἄλλα, Kat ἐκεῖνα εἴδει διαφέρει" καὶ yap ἡ φορὰ εἴδη ἔχει, av ἐκεῖνο ἔχῃ εἴδη ἐφ᾽ οὗ Kivetrar. ἔτι δὲ ἐὰν ᾧ, οἷον εἰ πόδες, βάδισις, εἰ δὲ πτέρυγες, πτῆσις; ἢ οὔ, ἀλλὰ τοῖς σχήμασιν ἡ φορὰ ἄλλη. ὥστε τὰ ἐν ἴσῳ χρόνῳ ταὐτὸ μέγεθος κινούμενα ἰσοταχῆ" τὸ αὐτὸ δὲ ἀδιάφορον εἴδει καὶ κινήσει ἀδιάφορον. ὥστε τοῦτο σκεπτέον, τίς διαφορὰ κινήσεως. καὶ σημαίνει 6 λόγος οὗτος ὅτι τὸ γένος οὐχ ἕν τι, ἀλλὰ παρὰ τοῦτο λανθάνει πολλά. εἰσί τε τῶν ὁμωνυμιῶν at μὲν πολὺ ἀπέχουσαι, ai δὲ ἔχουσαί τινα ὁμοιότητα, at δ᾽ ἐγγὺς ἢ γένει ἢ ἀναλογίᾳ, “ διὸ od δοκοῦσιν ὁμωνυμίαι εἶναι οὖσαι. Πότε οὖν ἕτερον τὸ εἶδος---ἐὰν ταὐτὸ ἐν ἄλλῳ, 7 dv ἄλλο ἐν ἄλλῳ; καὶ Tis ὅρος; ἢ TH κρινοῦμεν ὅτι ταὐτὸν τὸ λευκὸν καὶ TO γλυκὺ ἢ ἄλλο; ὅτι ἐν ἄλλῳ φαίνεται ἕτερον; ἢ ὅτι ὅλως οὐ ταὐτό; Περὶ δὲ δὴ ἀλλοιώσεως πῶς ἔσται ἰσοταχὴς α [Simplicius takes ἀλλὰ τοῖς σχήμασιν ἡ φορὰ ἄλλη to refer to differences between walking, flying, etc. ‘ We may dis- miss such distinctions as involving only a difference in the fashion of the movement’ (which would not affect the question whether the speeds were comparable).—C.]
» White and sweet were used as examples of ambiguous attributes above (248 b 20 ff.), and it was stated that not only has ‘sweet’ a different meaning as applied to water anes to ‘iba but it must have a subject of a different ind.—C.
¢ These questions of course are rhetorical. In either case the other-in-themselves and received-by-different-corre- spondents is supposed to be the obvious answer.
PHYSICS, VII. iv.
and the track in question are equivocal, for since the tracks are specifically different so are the movements that follow them. Are we to add that the instruments of locomotion establish specific differences in the motions themselves, walking with feet, and flying with wings? We may dismiss such distinctions and confine ourselves to the formation of the track,@ and say that the equal velocity means passing the same distance in equal time, only the ‘sameness’ must be specific in the case of the track and (consequently) in the case of the movement.
We must consider, then, what constitutes a specific difference in the case of motion. The above argument indicates that the genus is not indivisible, but that, besides the genus, there are a number of varieties concealed in it. When a term is equivocal, the senses covered by it may be widely removed, or they may have some resemblance, or they may, in fact or by the closeness of their analogies, draw so near to each other that the ambiguity of the term that includes them all, though very real, easily escapes our notice.
What constitutes a difference of kind then? Is it a difference in the several participants of the same nature, or rather a difference in that which is participated, to which the difference in the partici- pants corresponds? And how do we determine the existence of a specific difference? Or by what means are we to decide that ‘ white’ or ‘sweet’ is (specific- ally) the same or different (in a variety of cases)? Is it by noting only that they are manifested in differ- ent subjects, or because in both respects, in them- selves and their recipients alike, they are different? ¢ Turning, then, to modifications of quality, we must 5 10 15 ARISTOTLE ἑτέρᾳ ἑτέρα; εἰ δή ἐστι τὸ ὑγιάζεσθαι ἀλλοιοῦσθαι, μὴ μ “᾿Ν ἔστι τὸν μὲν ταχὺ τὸν δὲ βραδέως ἰαθῆναι, καὶ Ὁ Ὁ ἅμα τινάς: ὥστ᾽ ἔστιν ἀλλοίωσις ἰσοταχής, ἐν ἴσῳ yap χρόνῳ ἠλλοιώθη. ἀλλὰ τί ἠλλοιώθη; τὸ yap ἴσον οὐκ ἔσται ἐνταῦθα λεγόμενον, ἀλλ᾽ ὡς ἐν τῷ ποσῷ ἰσότης, ἐνταῦθα ὁμοιότης. ἀλλ᾽ ἔστω ? ἰσοταχὲς TO ἐν ἴσῳ χρόνῳ τὸ αὐτὸ μεταβάλλον. πότερον οὖν ἐν ᾧ τὸ πάθος ἢ τὸ πάθος δεῖ συμ- βάλλειν; ἐνταῦθα μὲν δὴ ὅτι ἡ ὑγίεια ἡ αὐτή, ἔστι λαβεῖν ὅτι οὔτε μᾶλλον οὔτε ἧττον ἀλλ ὁμοίως ὑπάρχει. ἐὰν δὲ τὸ πάθος ἄλλο 7, οἷον 9 fan ἀλλοιοῦται τὸ λευκαινόμενον καὶ τὸ ὑγιαζόμενον, τούτοις οὐδὲν τὸ αὐτὸ οὐδ᾽ ἴσον οὐδ᾽ ὅμοιον, ἧ ἤδη ταῦτα εἴδη ποιεῖ ἀλλοιώσεως, καὶ οὐκ ἔστι pia ὥσπερ οὐδ᾽ at φοραί. ὥστε ληπτέον πόσα εἴδη ἀλλοιώσεως καὶ πόσα φορᾶς. εἰ μὲν οὖν τὰ κινούμενα εἴδει διαφέρει ὧν εἰσιν al κινήσεις καθ᾽ αὑτὰ καὶ μὴ κατὰ συμβεβηκός, καὶ at κινήσεις εἴδει διοίσουσιν" εἰ δὲ γένει, γένει" εἰ δὲ ἀριθμῷ, ἀριθμῷ. ἀλλὰ δὴ πότερον εἰς τὸ πάθος δεῖ βλέψαι, ἐὰν ἢ α [The Oxf. Trans. takes τί as accusative and the question as meaning: What qualification are we to introduce into our definition of ‘ equal velocity ’ in the case of modification of quality corresponding to the ἡ equal distance (or ‘ same mag- nitude ’) covered ’ (249 a 19) in the case of locomotion? We must substitute for ὁ equality ° ‘ likeness ’ or specific identity.
> |‘ Them,’ 1.9. the two subjects just mentioned—the surface which is growing white, the body which is regaining health. The difference between whiteness and health is such as to imply a difference of kind between their respective subjects, which accordingly offer no ground of comparison, and the modifications they undergo must be specifically different.
.
PHYSICS, VII. rv.
ask under what conditions can velocities in change of quality be equal to each other? Now regaining health is a modification of quality, and it is possible for one man to gain health quickly and another slowly, and for two men to be doing so at the same time. So equal velocities of recovery must be possible, since the recoveries may occupy the same time. But what is the nature of that which is being modified?@ The term ‘equal’ cannot be used of a quale: in that category what corresponds to equality in the category of quantity is kkeness. Let us take ‘ equal velocity,’ then, to mean ‘making the same change in the same time.’ Are we then to compare the seat of the affections, or the affections themselves? In our present illustration, it is be- cause health (the affection) is the same that we are able to ascertain that it is present neither more nor less but in a like degree in the two subjects. But if the affection were different in each case— if one subject were whitened, for instance, and the other restored to health—there is no sameness or equality or likeness between them,® and this in itself establishes the two modifications as specifi- cally different, nor are they of one kind any more than the various species of locomotion we have examined. So we must ask how many species of qualitive modification there are, and how many of locomotion. If, then, the things which are directly and essentially (mot merely accidentally) the subjects of change or modification differ specifically, the movements or progress will differ specifically, and if generically generically, if individually indivi- dually. But is it enough to consider the nature of the affection and its identity or similarity if the modifi- ARISTOTLE 8490 ταὐτὸν ἢ ὅμοιον, εἰ ἰσοταχεῖς at ἀλλοιώσεις, ἢ εἰς τὸ ἀλλοιούμενον, οἷον εἰ τοῦ μὲν τοσονδὲ λελεύ- κανται τοῦ δὲ τοσονδί; ἢ εἰς ἄμφω: καὶ ἡ αὐτὴ μὲν ἢ ἄλλη τῷ πάθει, εἰ τὸ αὐτὸ (ἣὴ μὴ τὸ αὐτό), ἴση δ᾽ ἢ ἄνισος, εἰ ἐκεῖνο (ἴσον 7) ἄνισον.
Kat ἐπὶ γενέσεως δὲ καὶ φθορᾶς τὸ αὐτὸ σκεπτέον 30 ---πῶς ἰσοταχὴς ἡ γένεσις; εἰ ἐν ἴσῳ χρόνῳ τὸ αὐτὸ καὶ ἄτομον, οἷον ἄνθρωπος ἀλλὰ μὴ ζῶον' θάττων δέ, εἰ ἐν ἴσῳ ἕτερον. (οὐ γὰρ ἔχομέν τινα δύο, ἐν οἷς ἡ ἑτερότης ὡς ἡ ἀνομοιότης. καὶ εἰ ἔστιν ἀριθμὸς ἡ οὐσία, πλείων καὶ ἐλάττων ἀριθμὸς 25 ὁμοειδής" ἀλλ᾽ ἀνώνυμον τὸ κοινὸν καὶ τὸ ἑκάτερον, ὥσπερ τὸ πλεῖον πάθος ἢ τὸ ὑπερέχον μᾶλλον, τὸ δὲ ποσὸν μεῖζον).
1 [J have adopted the correction proposed by the Oxf. Trans. which renders: ‘That is to say, the alterations are the same or different according as the affections are the same or different, while they are equal or unequal according as the things altered are equal or unequal.’—C.]
2 [The stop after ἀνομοιότης might be removed: ‘ For we have not two terms to convey this difference in the way that unlikeness in quality is conveyed (by ‘‘more”’ and “ less ’’) or, on the theory that substance is number, a number (may be called) “‘ greater” or “‘less’’ than another of the same kind; but etc.’-—C.]
@ [‘ Equal’ or ‘ unequal ’—terms appropriate to the cate- ory of quantity—cannot be applied to the ‘ affection’ quality), but only to the subject, as having magnitude.—C. ] > Simplicius seems to be right in understanding this to mean if the embryo (of the same species) were brought to perfection in less time in one of the compared cases.
¢ [Jaeger, Aristoteles (1923) p. 313, sees a reference to the Platonic and Academic doctrine of substances as Ideal Numbers and refers to Met. M 7, 1080 b 37 ff.: the question whether the monads composing Ideal Numbers are com- PHYSICS, VII. rv.
cations it endures are to be regarded as of equal velocity, or are we also to take into consideration the seat or patient of the change, whether the same area has been whitened, for instance, in each case? Clearly both, so as to judge of identity by the affec- tion, but of equality by the area.% The same inquiry should be made as to genesis and perishing. What constitutes equal velocities of generation? If the times are equal, the things generated must be the same and of the same infima species, both men for instance, not merely both animals. It would be quicker if in the same time a different result were produced.’ (I use the terms ‘same’ and ‘different’ because we have no pair of terms to convey this difference in the case of a generation of a substance in the same way that in the case of alteration of quality unlike- ness is conveyed by the terms ‘more’ and ‘ less.’ If indeed being were number,° then there might be a ‘more’ or ‘less’ of number (z.e. being) of the same kind; but we have had to call it ‘otherness’ for want of a general word4 that can rank each being, as we rank the ‘more’ or ‘less’ (or rather excessive or defective) in quality and the ‘greater’ in quantity.)
parable (συμβληταί) or not, ete. If a number is to be com- parable (equal to, or greater or less than another) it must be of the same kind or order, ὁμοειδής. Cf. the use of this term ὦ [There is no special word (more precise than the wide expression ‘ difference’) corresponding to ‘ unlikeness’ in quality or ‘ inequality ’ in quantity to cover both relations (κοινόν), nor any word for each relation severally (τὸ ἑκάτερον), like ‘more’ and ‘less’ in quality, ‘greater’ and ‘smaller’ in quantity.—C. ] ARISTOTLE CHAPTER V ARGUMENT [This chapter states some simple principles of mechanics.
A force which can move a given load a certain distance in a certain time can move half the load either twice as far in the an time or the same distance in half the time (249 Ὁ 27- Also, if the load is reduced by half, half the force will suffice to move it the whole distance in the whole time (a 4-9).
Also, half the force can move the whole load, taken half at a time, half the distance in the whole time. But, if the load has to be moved as a whole, half the force wili not necessarily be able to move it half the distance or even to move it at all (a 9-19).
e49n27 "Eret δὲ TO κινοῦν κινεῖ TL GEL καὶ ἔν τινς Kal.
μέχρι του (λέγω δὲ τὸ μὲν ἔν τινι, ὅτι ἐν χρόνῳ, τὸ δὲ μέχρι του, ὅτι ποσόν τι μῆκος" ἀεὶ γὰρ ἅμα 80 κινεῖ καὶ κεκίνηκεν, ὥστε ποσόν TL ἔσται ὃ ἐκινήθη, καὶ ἐν ποσῷ), εἰ δὴ τὸ μὲν A τὸ κινοῦν, TO dé ΄ὮὌΝ > 250a τὸ κινούμενον, ὅσον δὲ κεκίνηται μῆκος TO [", ἐν ὅσῳ δὲ 6 χρόνος ep οὗ Δ, ἐν δὴ τῷ ἴσῳ χρόνῳ ἡ ἴση δύναμις ἡ ἐφ᾽ οὗ τὸ ἥμισυ τοῦ Β διπλασίαν τῆς Τ' κινήσει, τὴν δὲ τὸ Γ΄ ἐν τῷ ἡμίσει τοῦ A: οὕτω γὰρ ἀνάλογον ἔσται.
5 Kat εἰ ἡ αὐτὴ δύναμις τὸ αὐτὸ ἐν τῳδὶ TH χρόνῳ τοσήνδε κινεῖ καὶ τὴν ἡμίσειαν ἐν TH ἡμίσει, καὶ ἡ ἡμίσεια ἰσχὺς TO ἥμισυ κινήσει ἐν TH ἴσῳ χρόνῳ a ε A τὸ ἴσον. οἷον τῆς A δυνάμεως ἔστω ἡμίσεια ἡ TO α [Sir T. Heath (Greek Math. i. 345) remarks that the axiom which is regarded as containing the germ of the principle of virtual velocities is enunciated here and in the De caelo (301 Ὁ 4, 11) in the slightly different form: ‘ A smaller 10 15 ARISTOTLE EK καὶ τοῦ B τὸ Ζ ἥμισυ: ὁμοίως δὴ ἔχουσι καὶ ἀνάλογον ἡ ἰσχὺς πρὸς τὸ βάρος, ὥστε ἴσον ἐν ἴσῳ χρόνῳ κινήσουσιν.
Καὶ εἰ τὸ E τὸ Ζ κινεῖ ἐν τῷ Δ τὴν Τ', ἀναγ- καῖον᾽ ἐν τῷ ἴσῳ χρόνῳ τὸ ἐφ᾽ οὗ E τὸ διπλάσιον τοῦ Z κινεῖν τὴν ἡμίσειαν τῆς T. εἰ δὲ" τὸ A τὸ B® κινήσει ev τῷ A ὅση ἡ τὸ LT, τὸ ἥμισυ τοῦ (τὸ ἐφ᾽ οὗ Ἐ) τὸ B® οὐ κινήσει ἐν τῷ χρόνῳ ἐφ᾽ ᾧ A, οὐδ᾽ ἔν τινι τοῦ A, τὶ τῆς I ἀνάλογον πρὸς \ τὴν ὅλην τὴν T ὡς τὸ A πρὸς τὸ E. ὅλως yap, εἰ ἔτυχεν, οὐ κινήσει οὐδέν: οὐ yap εἰ ἡ ὅλη ἰσχὺς τοσήνδε ἐκίνησεν, ἡ ἡμίσεια οὐ κινήσει οὔτε ποσὴν οὔτ᾽ ἐν ὁποσῳοῦν" εἷς γὰρ ἂν κινοίη τὸ πλοῖον, εἴπερ ἥ τε τῶν νεωλκῶν τέμνεται ἰσχὺς εἰς τὸν Ἀ A ~ ἀριθμὸν καὶ TO μῆκος ὃ πάντες ἐκίνησαν.
1 [ἀνωγκαῖον EK, Simplicius (1105. 15 τῆς μὲν δυνάμεως πάλιν τὸ ἥμισν λαμβάνει, βάρος δὲ τὸ ἐξ ἀρχῆς, καὶ χρόνον ἐκεῖνον τὸν Δ. καὶ λέγει ὅτι τὸ ἥμισυ κινήσει τοῦ Τ' μήκους): οὐκ ἀνάγκη cett., Bekker, Prantl; Simplic. 1106. 1 records with approval a variant reading containing οὐκ. This sentence has been mis- understood by those who inserted οὐκ and by the editors who have followed them. With οὐκ the sentence becomes a duplicate of the next (in which εἰ δὴ 1s awkwardly substituted for εἰ 6¢); it ought to begin with εἰ δὲ or ἀλλ᾽ εἰ, not with καὶ el; and there seems to be no reason why ‘the same load’ = Simplicius calls it) should be called here τὸ διπλάσιον τοῦ and in the next sentence τὸ B. The loads intended are not the same, twice one not being for all purposes the same as two. τὸ διπλάσιον τοῦ Z means ‘twice F’ in the sense of two halves of B taken separately. As the next sentence says, the half-force E cannot cope with B as a whole, but our sentence says it can cope with it if allowed to move fist one half and then the other. If two colliers (A) carry two sacks PHYSICS, VII. v.
relation of the force A to the load B in the first, and accordingly the same distance (C) will be covered in the same time (D).
Also if E (4 A) will move F (4 B) over distance C in time D, it follows that in the same time Τὸ will move 2 F’s over half the distance C.* But if A will move B over the whole distance C in time D, half A (E) will not be able to move B, in time D or in any fraction of it, over a part of C bearing the same proportion to the whole of C that E bears to A. Because it may well happen that E cannot move B at all; for it does not follow that if the whole force could move it so far, half the force could move it either any particular distance or in any time what- ever; for if it were so, then a single man could haul the ship through a distance whose ratio to the whole distance is equal to the ratio of his individual force to the whole force of the gang.?
α [See critical note 1.—C.]
> | Literally, ‘ otherwise one man could move the ship, since both the force of the haulers and the distance over which all of them together make it move are divisible into the (same) number (of parts as there are men).’—C. ] of coal (B) a mile in an hour, one collier (E) cannot carry two sacks at once (B) at all, but he can carry two sacks, one at a time (2 F’s) half-a-mile within the hour. This principle —that the fractional force can deal with a multiple of its proper load if allowed to take it piecemeal—will explain ae misunderstood passage in Book VIII., 266 a 2 [δὲ EK: δὴ cett.—C.]
4 [ri τῆς Prantl (ef. Simplic. 1107. 2 μόριόν τι τοῦ Τ' δια- στήματοΞ), Oxf. Trans.: τις τῆς K: τῆς cett.—C.]
ARISTOTLE Διὰ τοῦτο 6 Ζήνωνος λόγος οὐκ ἀληθής, ὡς ψοφεῖ τῆς κέγχρου ὁτιοῦν μέρος" οὐδὲν γὰρ κωλύει μὴ κινεῖν τὸν ἀέρα ἐν μηδενὶ χρόνῳ τοῦτον ὃν ἐκίνησε πεσὼν ὃ ὅλος μέδιμνος. οὐδὲ δὴ τοσοῦτον μόριον, ὅσον ἂν κινήσειε τοῦ ὅλου, εἰ εἴη καθ᾽ αὑτὸ τοῦτο, οὐ κινεῖ: οὐδὲ yap οὐδὲν ἔστιν ἀλλ᾽ ἢ δυνάμει ἐν TH ὅλῳ. i δὲ τὰ δύο, ἑκάτερον δὲ τῶνδε ἑκάτερον κινεῖ τοσόνδε ἐν τοσῷδε, καὶ συντιθέμεναι αἱ δυνάμεις τὸ σύνθετον ἐκ τῶν βαρῶν τὸ ἴσον κινήσουσι μῆκος καὶ ἐν ἴσῳ χρόνῳ" ἀνάλογον γάρ. Ap’ οὖν οὕτω καὶ ἐπ᾽ ἀλλοιώσεως Kal ἐπ αὐξήσεως; τὶ μὲν yap τὸ αὖξον, τὶ δὲ τὸ αὐξανό- 30 μενον, ἐν ποσῷ δὲ χρόνῳ Kal ποσὸν TO μὲν αὔξει τὸ δὲ αὐξάνεται. καὶ τὸ ἀλλοιοῦν καὶ τὸ ἀλλοιού- 250b μενον ὡσαύτως, τὶ Kal ποσὸν κατὰ TO μᾶλλον καὶ τὸ ἧττον ἠλλοίωται καὶ ἐν ποσῷ χρόνῳ---ἐν διπλασίῳ διπλάσιον (ἢ ἐν ἡμίσει ἡμισύλ,Σ καὶ τὸ 1 [Fort. εἰ δὲ τὰ {κινοῦντα δύο. Cf. Philop. 881. 18 εἰ δύο τινὰ εἶεν κινοῦντα καὶ ἑκάτερον αὐτῶν κινῇ, KTA.—C.]
5 [In the mss. this clause stands after τὸ δ᾽ ἥμισυ ἐν ἡμίσει χρόνῳ. In this sentence τὸ διπλάσιον and τὸ ἥμισυ mean “twice the thing that is altered’ and ‘half the thing that is altered’; while διπλάσιον and ἥμισυ are adverbial with ἀλλοιοῦται understood, and mean twice and half the amount of change. (Here I differ from the Oxford Translation, which treats ἥμισυ as sometimes equivalent to τὸ ἥμισυ.) The last clause ἢ ἐν ἴσῳ διπλάσιον must have τὸ ἥμισυ for its subject. Either τὸ ἥμισυν must be inserted in it, or the clause which precedes it in the mss., ἢ ἐν ἡμίσει ἥμισυ (the subject of which is not τὸ ἥμισυ), must be transposed where I have put it, so as to allow the τὸ ἥμισυ in τὸ δ᾽ ἥμισυ ἐν ἡμίσει χρόνῳ to serve as subject to the last clause.—C.]
6 (It appears from Simplicius 1108. 18 that Aristotle is here referring not to Zeno’s own writings but to some early dialogue (Diels suggests the Φυσικός of Alcidamas, Vors.4 19 a 29) in which Zeno was represented as arguing with Protagoras.—C.
PHYSICS, VII. v.
And in this lies the fallacy of Zeno’s * contention that every grain of millet must make a sound as it falls (if the whole measure is to doso). For it may well be that in no period of time could the one grain make such astirin the air ὃ asthe whole measure does. Nor need it be able, if alone, to effect that portion of the total movement which may be assigned to it in accordance with its proportion to the whole mass°; for it cannot be regarded, except potentially, as having any several action in the total movement effected.
On the other hand, if two separate agents can each of them move one of two loads in so much time, then if united they would move the combined load the same distance in the same time; for the proportions would hold.
And it is the same with qualitive modifications and with growth. For there is something that causes the growth and something that is made to grow, and the process takes so much time, and the growth effected and acquired is somuch. So too with the qualitive modification and the quality modified, for a certain “so much,’ as measured by ‘more and less,’ is modi- fied, andin ‘so much’ time. Thus if A alters Btoadegree Cintime D, then Ay, δ es 2C, 2D > (Literally, ‘ move that air that the whole bushel moves’ i.e. such an amount of air as we may suppose necessary to cause the whole sound.—C. | ¢ [Just as one out of 100 ship-haulers cannot by himself move the ship zj,th of the distance. So Simplicius (1109. 2); but the sentence can more easily be construed to mean: ‘ In- deed (when it forms a part of the whole bushel) it does not even move such a part of the whole amount of air as it would move if it were by itself; for no part so much as exists (as a distinct moving agent) in the whole otherwise than potenti- ally.” Cf. Oxf. Trans.—C.]
ARISTOTLE 250b διπλάσιον ἐν διπλασίῳ, TO δ᾽ ἥμισυ ev ἡμίσει χρόνῳ ἢ ἐν ἴσῳ διπλάσιον. Et δὲ τὸ ἀλλοιοῦν ἢ αὖξόν re’ τοσόνδε ἐν τῷ 56 Ἅ af aN LAA ag \ \ ΦΨ 2 € ὔ 5 τοσῷδε ἢ αὔξει ἢ ἀλλοιοῖ" (καὶ τὸ ἥμισυ ἐν ἡμίσει ἢ ἐν ἡμίσει ἥμισυ), οὐκ ἀνάγκη (καὶ τὸ ἥμισυ ἐν διπλασίῳ), ἀλλ᾽ οὐδέν, εἰ ἔτυχεν, ἀλλοιώσει ἢ αὐξήσει, ὥσπερ καὶ ἐπὶ τοῦ βάρους.
1 [τι: τὸ MSS. τὸ τοσόνδε Should mean ‘an object (area or volume) of a certain magnitude,’ but some mention of ‘a certain amount’ of change seems needed; ef. 250 a 25. Perhaps τὸ τοσόνδε <roodvde>.—C. ] 2 [The ms. reading ἀλλοιοῖ, οὐκ ἀνάγκη καὶ τὸ ἥμισυ ἐν ἡμίσει καὶ ἐν ἡμίσει τὸ ἥμισυ, ἀλλ᾽ οὐδέν, κτλ. is nonsense. In the clause καὶ τὸ ἥμισυ ἐν ἡμίσει, where it stands in the μ88., if τὸ ἥμισυ means τὸ ἥμισν τοῦ ἀλλοιοῦντος (+ A), it is nonsense, for no one could expect half the force to do the work in half the time; if it means τὸ ἥμισυ τοῦ ἀλλοιουμένου ($ B), the statement is false, for then the subject is τὸ ἀλλοιοῦν (A) and A will change } B in half the time. The other clause καὶ ἐν ἡμίσει τὸ ἥμισυ is a mere repetition, which should in any case be corrected from Simplicius to ἢ ἐν ἡμίσει ἥμισυ.
The remedy adopted is based on Simplicius’s paraphrase (1111. 18): οὐκέτι μέντοι εἰ τὸ τοσόνδε ὑπὸ τοῦ τοσοῦδε ἠλλοίωται τήνδε τινὰ τὴν ἀλλοίωσιν ἢ ηὔξηται τήνδε τὴν αὔξησιν, ἀνάγκη καὶ τὸ ἥμισυ ἐν ἡμίσει ἢ (sic) ἐν ἡμίσει ἥμισυ (sic) ἀλλοιοῦσθαι ἢ αὔξεσθαι ὑπὸ τῆς ἡμισείας δυνάμεως ἢ ὑπὸ τῆς ἡμισείας τὸ ὅλον ἐν διπλασίῳ χρόνῳ ἀλλοιωθήσεται. οὐδὲ yap τοῦ ὅλου πάντως κινητικὴ ἡ ἡμίσεια δύναμις, GAN’ εἰ ἔτυχε τοσαύτη ἐστὶν ἡ ἡμίσεια ws μὴ ἰσχύειν ἔτι ἀλλοιοῦν ὅλως ἢ αὔξειν. This suggests that Simplicius read: οὐκ ἀνάγκη καὶ τὸ ἥμισυ (τι ὑπὸ τῆς ἡμισείας δυνάμεως) τὸ ἥμισυ ἐν ἡμίσει ἢ ἐν ἡμισει ἥμισυ, ἢ τὸ ἥμισυ (= ὑπὸ τῆς ἡμισείας) ἐν διπλασίῳ, and understood it to mean ‘It does not follow that 4 A will change + B by amount C in time ᾧ D (nonsense) or change B by amount $ C in time 3 Ὁ cna otherwise (ἢ) 4 A will change B by amount C in 2 D; (and this will not do,) for the half-force may PHYSICS, VII. v.
or Alters B toa degree $C in time Ὁ D and A, 2B Pe ον 8 while A, 8 τ Cc 4, ¢D or A 1B ss OC sy De If, on the other hand, there is a force A causing alteration or growth, and if A alters Btoadegree Cintime D it does not necessarily follow that 4 A willalter B to a degree C in 2 D,? but it may well happen that 4 A will effect no change or growth whatever, just as in the case of the load. ¢ [With the transposition made in the text the sentence becomes logical, and the use of ἤ (not «ai, twice) correctly indicates alternative effects that may happen to the same subject. To make the statement symmetrical and complete, it would be necessary to add (after καὶ τὸ διπλάσιον ἐν δι- πλασίῳ) ἢ ἐν ἴσῳ ἥμισυ, t.e.°2 B will be altered ᾧ C in Ὁ. But the omission may be Aristotle’s.—C. δ [Nor, it might be added, will } A necessarily alter B by amount $C in time D (=4 ἐν ἔσῳ ἥμισυ). (Cf. above,