eis TO ev is apparently to be supplied with éuaurrovays, being under- stood from 6 ad’ évos durdAacalopevos, while with di 8& rév wepurrav we must supply in thought the eis tov dptiov of ll. 4,5. The main opposition is that between generation of numbers by addition and by multiplication, the latter being subdivided. Accordingly Aristotle says Gd d€ THS ev Svados, Meaning to continue with ray dé repirtav. But by an oversight he continues with wot d¢ tév qwepittav.
Alexander offers a more elaborate classification, which doubtless preserves some real information about the Pythagorean and Platonic arithmetic (cf. Heath, Gk. Math. i. 71-74). According to him every genesis of number is
(1) dpridkis dpa (powers of 2, = (2 a) above), or
(2) dpriomépiacos (products of an odd number and 2), or
(3) mepuroaprios (products of an odd number and 4 or a higher power of 2), or
(4) apy kal dovvberos (prime numbers), or
(5) Sevrépa kat avvOeros (composite odd numbers), or
(6) kal?’ éavrnv pev devrépa Kat ovvOeros mpos adAov 8é ‘pity Ka dovvOeros (pairs of composite numbers which are prime to one another). ;
Alexander supposes that (1), (2 2), and (2 2) of Aristotle’s classifica- tion are identical with (4), (1), and (2) of his own, and that Aristotle omits the rest dia Bpaxvdoyiay (769. 21). But it is evident that a com- plete classification is necessary to Aristotle’s purpose. Aristotle’s (1) includes Alexander’s (4) and (5); his (2 4) includes Alexander’s (2) and (3); and Alexander’s (6) has no proper place in the classification, since it depends on a relation between two numbers, not on a quality of one. -
Q. ote Kata tiv Odow évBdxeTat, i.e. it is incompatible with the notion of the Idea as a principle of limit; ote kata Aédyoy, i. e. it is unreasonable in itself, since it implies the existence of an actual infinite.
10, tdtrovci y oltw tds iS€as. The manuscript reading (rérrovor 5 otrw ras ideas) can hardly mean, as Alexander supposes, ‘ but they
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limit the series of ideal numbers to 10’. Since this is not mentioned till 1. 12, it cannot be what otrw means here. The word would have to refer to 1, 7, and mean ‘but they conceive of the Ideas as Ideas of something, and of the numbers as being Ideas’. Schwegler’s emenda- tion is undoubtedly right ; raérrovoi y' crv. = ‘i.e. for those who arrange the Ideas as they do’, i.e. identifying each Idea with a finite number.
-12, ei péxpe tis Sexddos 6 dpiOuds. Cf. ll. 25-34, N. 1088) 10, A. 1073219 ot A€yovtes ideas wept... THY GpiOpGv dre pev Os Tept dmeipwy A€yovow Ste dé ds pexpr THS dexddos dpirpevwv. In Phys. -206> 32 the doctrine is ascribed to Plato by name: péxpu yap dexados move’ Tov dpiOudy. Speusippus is probably also referred to (ch Z. 1028> 21 n.). The origin of the view is of course to be found in the fact that the Greeks used a decimal system and in the reverence paid by the Pythagoreans to the number 10; cf. A. 986*8 and Philolaus fr. rr Diels. to was the sum of the first four numbers, the terpaxtvs, Which had a special significance because they were the principles of the point, the straight line, the triangle, and the tetra- hedron respectively.
15. Bz. (dnd. Ar. 12575) takes aité as predicate ; ‘each number up to 10 is a thing-itself (an Idea)’; cf. Zop. 162° 28. It seems better to take aird Exaoros dpiOds together = 6 cidytixds apiOuds, as Alexander does (770. 23), and péxps dexddos as predicate. atrd éxacros apiOpds = ‘the series of numbers which are the several things-themselves (the Ideas of the several things)’. For atré exaoros cf. Top. 162° 24, £. NV. 1096? 35.
16. tav év tovTois dpOuav is usually interpreted as ‘the numbers within ‘these limits’, i.e. between 1 and ro. But on this view what is the point of dAX’ pus (1. 17)? That suggests that in spite of there being a /arge variety of numbers to choose the Ideas of different kinds of animals from, there would not be enough. Now the notion of numbers contained in other numbers is clearly in Aristotle’s mind (cf. Il. 18, 19 and notes) and is expressed similarly by év. May it not be that Aristotle uses év rovrous in a double sense? ‘The Idea of horse © must be one of the numbers contained in these, i.e. either one of the numbers between 1 and ro, or one of the numbers contained in those between 1 and 10,’
18-21. There is little to be said for Christ’s transposition of this section to]. 25. ovrws refers quite as naturally to |. 14 as it would to 1. 22 (6 ek rév ovpBAyrov povddurv).
18. at adda tpiddes. Alexander explains this (770. 30) as ai tpuades THs avtoeEddos Kal trav NowrHv, and similarly Bz. thinks the 3’s included in the other ideal numbers (cf. 10824 2, 28, » 13) are meant. Against this Robin argues (p. 351, n. 7) that on the view here criticized the ideal numbers are limited to ten, and the ‘ other 3’s’ in these will be only 14 in number (1 in 4, 1 in 5, 2 in 6, 2 in 7, 2 in 8, 3 ing, 3 in 10), so that the conclusion depo. éoovrar avOpwrot (1. 20) will not follow. He therefore supposes Aristotle to be now taking account of mathematical numbers, which are not limited to 10, and saying that
each 3 contained in them, since it is like the ideal 3, will be some sort -of a man, even if not an ideal man. We may either suppose this, or suppose Aristotle to be taking account of a further complication within the series of the ideal numbers. Besides the 14 3’s of which Robin takes account, there will be the 3 which is in the 4 which is in the 5, the 3 in the 4 which is in the 6, the 3 in the 5 which is in the 6, the 3 in the 4 in the 5 in the 6, &c. A list which may fairly be called dreipov is thus produced.
IQ. oporar yap at év toils adtois dpiOpots. Bz. supposes idéar to be the noun implied by ai, and takes the phrase to mean ‘the Ideas con- sisting in identical numbers’. But zpuddes is the only word that can be supplied ; and further Bz.’s interpretation assumes that the dAdo tpuddes are Ideas, which Aristotle expressly leaves uncertain (Il. 20, 21). It seems better to suppose, with Robin (p. 352), that Aristotle means that the 3 which is in the 4 itself is like the 3 which is in the 4 which is in the 6, and the 3 which is in the 6 itself is like the 3 which is in the 6 which is in the 7 itself, and so on. Yet even this interpreta- tion is not quite satisfactory, since to justify ai dAAa (ad/ the other) tpiddes Aristotle ought also to mean that the 3 in the 4 itself is like the 3 in the 6 itself. But probably Aristotle overlooked this point.
20-21. ‘Ifeach 3 is an Idea, each of the numbers will be Man Him- self.’
25. The assignment of numbers to the different Ideas by Aristotle is arbitrary ; itis quite unnecessary to read tpids for duds with Christ to bring the sentence into conformity with Jl. 14, 18.
27-29. Bz. thinks this is an interpolation, belonging to the criticism not of ideal numbers but of Ideas in general (cf. 1080%2-8). The passage is, however, interpreted by Alexander and Syrianus without any suspicion of its spuriousness, and it seems quite possible to connect it with what precedes, if we interpret «id as meaning ideal numbers, which in view of the repeated identification of Ideas with numbers we are entitled to do. Aristotle has just referred (il. 25-27) to the arbitrary assertion of the existence of Ideas of numbers up to ro, and the arbitrary denial of their existence beyond that point. Here he points to a similarly arbitrary distinction. ‘Forms are introduced to explain being and becoming.” Yet some things (negations 10794 9, relations 1079° 12, manufactured objects 1080 5) are and become with- out being supposed to have Forms answering tothem. Why have they not Forms? ‘The fact that the Platonists can dispense with Forms in these cases shows that Forms are not the causes of being and becoming.’
30. padddv tr ov. It seems pretty clear that Alexander and Syrianus had the same reading as our manuscripts. Alexander interprets it as meaning kal radra 70 ev Kar’ avrovs paAdév Tu ov KTA.; SO also Syrianus (Alexander may have read in the next clause xa/ for kairo). Bz.’s proposal to read «i 6 dpuOpos prexpr THs Sexddos, parAdv Te dv Td Ev Kal eldos xtA. has not the authority of the Greek commentators; and the accusative absolute is not probable. The interpretations of Alexa: and Syrianus do not commend themselves. According to Alexande
2573-2 Gre:
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argument is: ‘If the One is both the Form of the ro and ungenerated, while the ro has come into being, there will be an 11 whose Form is the One and whose matter'is the decad’, According to Syrianus the One in question is the One which is the formal principle of number, and what it is in relation to all the numbers, the ideal ro is in relation to the other tens, the hundreds, and the thousands, for which reason it was called Sevrepwdovpéva povas. The meaning seems to be: ‘Further, it is paradoxical if the number series up to ro is more of an entity and a Form than the ro itself; to this we may object that there is no generation of the series as a unity, while there is of the 10. Yet they try to speak as if the number series up to 10 were complete’, Certain Platonists may have said something (we do not know what) to justify Aristotle in describing them as holding the series up to ro to be more of an entity than the 10 itself; once grant this and Aristotle’s objection becomes plain. He objects that, as the Platonic theory only describes the origin of the numbers severally and not of the series ws évds, the series cannot form a true entity or Form.
32. Ta émdpeva, ‘the derivative entities’.
33. To kevov kT. Alexander explains that the space between the even numbers 2, 4, 6, 8, or again between the odd numbers 3, 5, 7, 9 was the Idea or pattern of the void (this may be an inference from Phys. 213° 24 70 yap kevov duopilew thy piow airar, sc. Tov apiOpar, but it is of the Pythagoreans that Aristotle says this); that ‘2, 4, (6), 8’ was the pattern of arithmetical and ‘2, 3, 6, 9’ the pattern of geometrical proportion; that the number 1 was the Idea of oddness ; while movement and good were derived from the One, rest and evil from the indefinite dyad. Thus he takes ra dAAa (1. 35) to refer to ro Kevov, THY dvaoylav, TO mepitTov, TA GAAa Ta ToLadTa. On the other hand Theophrastus (JZef, 312. 18—313. 3 Br. = fr. xii. 11 fin., 12 Wimm.) says that the Platonists derived place, she vord, the infinite from the indefinite dyad, and certain other things, e.g. soul, from the numbers and the One. Robin accordingly (p. 317) takes 70 Kevov, dvadoyia, To mepitrév, as well as xivyows, ordows, dyabdv, Kakov to have been derived from the dpyai (the One and the indefinite dyad), and thinks that 7a dAAa (1. 35) is left here without illustration but means what Theophrastus describes as yy) kal GAN drra. With Theophrastus’ statement that the void was derived from the indefinite dyad cf. Phys. 209> 11 IAdrov tiv vAnv kal thy xdpav taiTd dyow etva. Zhe odd is actually described in 1. 36 as identified by the Platonists with one of the dpyai, the One, In this the Platonists followed the Pythagoreans, who described the formal principle indifferently as the limit and the odd. The best explanation of the reference to proportion is furnished by Syrianus, who points out that instances of all the three fundamental évaAoyia. ean be found without going beyond the number ro: arithmetical dvaAoyia, e.g. 1, 2, 33 geometrical, e.g. 1, 2,4; harmonic, e.g. 2, 3, 6. For the derivation of movement from the indefinite dyad cf. K. 1066 11, where we are told that some thinkers describe movement as érepéryra Kal dvicdryta Kal Td
pa ov (which = 70. péya Kal 7d puxpov Phys. 1927), and A. 992? 7. Eudemus also says that Plato identified movement with the great and small (ap. Simpl. PAys. 431. 6, 13, p. 41. 18, 42. 8 Spengel). For the reference of good and evel to the One and the indefinite dyad respectively cf. A. 9884 14.