The supposition that number comes from its principles ds e& evv- mapxovrwy is sufficiently refuted by the answer that it is only things that are generated (whether naturally or artificially) that are é€ évu- mapxovrwv—while the numbers are eternal (cf. 1088 14-28). as did o7repparos is one form of as éx py evuTrapyYovTwy. ;
GAN obx ofdy Te Tod Adiatpérou Ti dmeAOetv. Bz. interprets thus: “To explain growth from seed, something must be supposed to come off from the seed; but nothing can come off from the indivisible One’. But it is no part of Aristotle’s theory to suppose that in generation something comes off from the seed; rather the seed comes off from the male parent (for this use of dwedOeciv cf. Meteor. 4669, P. A. 641° 34, G.A. 721 12 ef passim).
33. GAN ds ek Tod évaytiou py Gmopevovtos; Aristotle now passes to another case of production é« pi) évurdpxovros. Production from a contrary which does not persist (cf. a. 994% 24, 30) is dAAoiwors, not yeveots amdh ; for yéveois arAF is yeveots Kara. THY ovciay, and an ovcia has no contrary (1087 3 n.); yéveows dAf is peraBodr Kat’ dvribacw, from the absence of a substance to its presence, not from one contrary to another (PAys. 2252 12-17), and it involves no persisting sensible substratum (De Gen. e¢ Corr. 319» 14, 33), while the kind of production here in question does involve such a substratum (109 2° 34).
34-3. The argument against the suggestion that number comes from a contrary which does not persist runs thus: ‘What comes from its contrary presupposes also something that persists. Now number
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is represented as coming from contraries. Therefore there must be something else, from which, as well as from one contrary, number is produced’. The reasoning is evidently fallacious. The major premise states that when A is produced from zs contrary B, a substratum C is presupposed. The minor states that number is according to the Platonists produced from /wo contraries. Production of a thing from its contrary, and production of a thing from.two contraries, are quite different, and there is a fallacy of four terms. Aristotle perhaps means to say that instead of deriving number from the One and plurality the Platonists should have derived it (since it is itself plurality) from the contrary of plurality, viz. the One, and from something which can be the substratum at one time of oneness, at another of plurality, as the change from white to black implies a substratum which can be either ; but if this be his meaning, he does not express it clearly.
35- 6 pév. In the light of 1091 32 and A. 1072? 31 it is clear that this means Speusippus; cf. M, 10859 n.,5n, Plutarch (De An. Procr. ii. 1. 2, 1012 D £) tells us that Xenocraées spoke of the material principle as plurality, but of this we cannot be so sure. 6 8€(?1) means Plato,
b4. dca e€ évaytiwy contains the ambiguity commented on in the note on *® 343; it might mean things of which each is produced from its contrary, or things each of which is produced from two con- traries. Kav ék wavrds 9, ‘even if all the contrary is used up’ (which refers to this case alone, not to ois éorw évayria) shows that the former is meant. The contrary which has been used up in making a thing is conceived of as still potentially present in it (évurdpxov |. 6) and capable of destroying it, no less than a contrary which is outside the thing (1) évurdpxov).
7. otov 7 veikos 76 piypa, in Empedocles’ system.
(E) Mumbers as causes of other things (ch. 5. 1092” 8—6, 1093 29).
8. We are not told how numbers are the causes of substances, whether (1) as boundaries (as points are of spatial magnitudes, or as Eurytus determined the number of each thing by counting the pebbles he used in tracing its outline), or (2) because harmony, man, and every- thing else is a ratio of numbers. ,
15. But (1) how can attributes, like white, be numbers? (2) The ratio is the substance of a thing ; the number is merely matter. ‘The number is always a number of something, of portions of fire or earth or of units; the substance is ‘so much to so much ’, i. e.a ratio of mixture of numbers.
23. Thus number is neither efficient, material, formal, nor final cause of things.
26. What is the good that comes from the fact that a mixture is expressible in numbers? (1) It is more important that honey-water should not be too strong than that its elements be in any particular ratio,
30. (2) The ratios of mixture involve not numbers merely but the addition of numbers, not ‘thrice two’ but ‘three of one to two of another’, while in multiplication the genus must be the same.
1093°1I. (3) Ifallthings share in number, (a) it does not follow that a thing’s number is its cause, (4) many things have the same number and will therefore, on the theory in question, be the same.
1g. There are seven Pleiads and there were seven against Thebes, but the nature of the number seven is not the cause of their being seven,
20. They say there are only the three double letters, BWZ, because there are just three concords ; but the number is in either case arbitrary. We are reminded of the methods of the old Homeric scholars. Other numerical comparisons which the Platonists make are equally frivolous,
b, The lauded characteristics of numbers are not causes in any of the senses of ‘ cause’ ; but these thinkers do show in a sense that good- ness belongs tothe odd, the straight, &c. There is a sort of correspon- dence between the straight in length, the even in breadth, the odd in
number, the white in colour. at, (4) It is not the ideal numbers that are the causes of harmonic relations and so forth (for equal ideal numbers are different in kind), so that this affords no reason for believing in Ideas.
24. These difficulties show that mathematical objects are not separate from sensible objects, and that the first principles are not those which these thinkers put forward.
1092» 8—109329. Aristotle now passes from the alleged genesis of numbers to the alleged genesis of things out of numbers. The following considerations show that he has in mind the Pythagoreans and perhaps also the Platonists who most resembled them :
(1) The mention of Eurytus.
(2) The reference to the connexions established between numbers and movements of sun and moon, and the periods of animal life (1093* 4-6, cf. A. 986% 3-8).
(3) The reference to the significance of the number 7 (1093* 13).
(4) The reference to the two cvororxias (1093° T1-14).
It is only at 1093 21 that Aristotle turns to the Platonic theory of
Idea-numbers. 8-15. Mr. Cornford suggests with much probability (Class. Quart.
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xvii. ro f.) that the second of the alternatives here mentioned, the view that ‘things embody or represent numbers, not are numbers ’, is the origi- nal Pythagorean doctrine; and that the first alternative, ‘the crude materialistic view... that things ave numbers, and numbers consist of monads, which are the terms or boundary-stones (6po.) marking out the void “ field” (yépa) in the geometrical patterns of numbers “figured” by pebbles’, is-a later, fifth-century doctrine. Aristotle’s doubt as to the meaning of the Pythagoreans may well be due to his not having distinguished successive phases of their doctrine. Cf. M. 1080? 16 n.
10. We may infer from Theophr. Ae/, 11 Wimm, = 312. 15 Br. that the source of Aristotle’s information about Eurytus was Archytas. Eurytus belongs to the beginning of the fourth century; he was a disciple of Philolaus. Alexander tells us that his method was to sketch the outline of, e.g., a man with coloured pebbles, and then to say that the number of pebbles he had used, say 250, was the number ofman. ‘This is a travesty of the method of limits according to which the early Pythagoreans represented the line by the number 2 (the number of points required to determine it), the surface by 3, the solid by 4 (cf. Z. 10368 n.).. The same process was as it were worked backwards when the numbers were ‘reduced to the forms of triangle ane square’ (l. 12). 3 (.*.), 6 (.:) and all numbers of the form “** were triangular (cf. Plut. Plaz. Quaest, 2.1003 F). Nicomachus of Gerasa (Introd. Arithm. ii. 8-11) and Theo of Smyrna (pp. 31-33 Hiller) represent numbers by a’s arranged in various geometrical patterns:
I 2 3 4 5 6 9
a aa a aa a (9 ee Aas 3 aaa
aad aa aa a aa a ae
ag aad
‘Square’ and ‘cube’ survive as names for kinds of numbers; ‘linear’, ‘plane’, ‘solid’, ‘ triangular’, ‘ oblong’, ‘ pentagonal ’, and other names which belong to the same geometrizing order of thought have passed out of current use, but, as Prof. Burnet observes, we still represent numbers in this Pythagorean way on dice and dominoes, and we still call numbers figures. On ‘figured numbers’ cf. Heath, Hucld, vol. ii. 288 f., Gk. Math. i. 76-84, Burnet, L. G. P.§ 47, G.P. 53, Iamblichus, Zxfrod. p. 56. 26 ff. Pistelli. According to Lucian (Biwy mpacis 4) the ‘triangular numbers’ were recognized by Pythagoras himself,
13. utd is surprising when the instances given have been man and horse. Christ conjectures Cgwv xal durdv from Alexander (827. 26), but it is by no means clear that Alexander read this (cf. 826, 35), and it seems more likely that Aristotle uses durdv in its older and wider sense of ‘living being’, which is found in Plato (Soph. 233 8 8, Rep.
401 44, Zim.goAa6). Aristotle may be quoting from Archytas’ account of Eurytus,
15-16. 7a 5é 8h wdOn. . . Oepudy. This is an objection to the first suggested mode of treating numbers as causes of things (Il. 9-13); you can make an outline of a man, but how are you to sketch the outline of a aOos?
16-17. om 8€... SHAov is an objection to the second suggested mode (Jl. 13-15); if harmony is a Adyos dpufyav, the numbers are merely the matter, the ratio is the essence.
18. 68 dp.Opds dAy is in verbal contradiction with ovre vAy |. 24, and Schwegler would read tags (cf. Al. 827. 40 6 d& dpibuds... 7d moaov éort THS ExdaTou VAys). Aristotle’s view, however, is that ap. .0s cwparixds (1. 22) (i.e. amounts of certain simpler forms of matter, numerically determined) is the vAy of a compound, though dpibpuds povaducos (1. 20) is not the vAy of any material thing but only the measure of its constituents; so that both statements are true, though of number in different senses.
19. Tpia mupds ys S€ SUo. This suggestion is clearly framed on the analogy of Empedocles’ analysis of bone—
tas S00 Tav éxrd pepewy Ade Nyoridos aiyAys, téccapa 8 “Hdaicrowo: ra 8 doréa AevKa yévovto (De An. 4107 5).
GpiOpos ... mupivos 4 ynivos. Cf. the dpiOuot pydtra, piadtras, numbers of apples (or of sheep), of bowls, &c,, which Geminus (cf. Procl. 7 Eucl. i, p. 40. 2-5) and the scholiast on the Charmides (165 £) describe as studied by Aoyrrixy in distinction from dpibpntixy (Heath, Atst. of Gk. Math. i. 14, ii. 442).
25. If number is in no sense the cause of things, how is it related to them? Aristotle nowhere gives a positive theory of number (the nearest approach is in M. 3), but his answer might be that number is an oixetov 7aOos of their matter.
27. év edhoyiot» means not, as Alexander says, éy dpriy, i.e. in a ratio like 1:2 or 1: 4, but ‘in an easily reckoned ratio’ (illustrated in De Sensu 439° 25—4402 6 by 3:2 and3:4). It excludes (r) ratios of which either term is an irrational, and perhaps also (2) ratios which cannot be expressed in terms of numbers within the series 1-10,
28. év mepitta, in a ratio like that of 1:3 (Alexander) or possibly like that of 2:3 (~:2+41). For the virtues of the odd number according to the Pythagoreans cf. A. 9864 18 n.
tpis tpta. The meaning of this difficult phrase is clear from 1. 32. Aristotle supposes the Pythagoreans, when they say tpis dvo, to mean ‘three to two’. When they gave zpis zpéa, then, as the recipe for peAtkpatov, they meant that if you take three parts of honey you must take three parts of the other ingredient (milk in Homeric times, later water). Alexander’s mention of a third ingredient, saffron, seems due to a misinterpretation of zpis tpia; it is contrary to his statements about pAikparov elsewhere and to those of other writers. Cf, Columella xii. 12.
29. év ob8é Adyw. Aristotle does not mean that the constituents
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could be in no ratio; he means that the particular ratio does not matter provided there is enough water.