(2) That it is something analogous to 7A7Gos, i.e. something which is to spatial magnitudes what multitude is to numbers (10854 33). wAnGos is mentioned in various places as the material principle of number according to some Platonists ( 5, N. 10876, 27, 1091 31, 10928 28, 35). A comparison of 1091) 30-35 with 10914 29-» 3, and with A. 1072 30 (where he is mentioned by name), makes it pretty certain that Speusippus is the thinker referred to. Cf. Al. 823. 12. Plut. De An. Procr.ii, 1, 2. 1012 DE, ascribes the view to Xenocrates, but his testimony is of less worth.
13-14. ‘As regards the principle in such objects which answers to the One (i.e. which is for geometrical objects what the One is for numbers), different thinkers hold different views.’ Al. 777. 17 distinguishes two views. (1) Some thought that the ideal numbers were the forms of geometrical objects—two the form of the line, three of the plane, four of the solid. (2) Others thought that the One was their form. Both views are suggested in B. roo1? 24.
(1) This view is mentioned in N. 1ogo} 22, Z. 103613. It may be that the holder of this view was Xenocrates, for in Z. 1028» 24, after saying that Speusippus severed the various classes of being from each other, Aristotle continues with the words évou dé ra pev td Kat rovs dpOpors Tiv aitny exew dact pic (this shows that Xenocrates is
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in question, cf. M. 1076? 20 n.), 7a. dé dAAa exomeva, ypappas Kat eizeda., péxpt pos THY TOD otpavod ovciay Kat Ta aicOyrd. I.e. these thinkers linked up the various classes of entity, and made ypappas kal érireda dependent on the numbers. Cf. Theophr. Jef. 313. 4 Br. = fr. xil. 12 Wimm. The view of Zeller (ii. 1.4 949, n. 2) that Plato was the holder of this view seems less probable. “Al. 777. 16, Syr. 154.9 refer the view to Plato, but we have seen reason to believe that they knew little about early Platonism. Cf, Robin, pp. 295-298.
We have no further information about view (2), but (3) a third view is expressed in |. 32, the view that the formal principle was the point, considered as analogous to the One. Cf. 27. The persons who held this view are identified by Aristotle with those who took the material principle to be otoy wAOos, and we saw (l. 9 n.) that Speusippus is meant. It could not be either Plato or Xenocrates that is referred to, for they did not believe in points (for Plato cf A. 992 20).
16. dmohedupéva te is caught up by rairé re |. 20. The argument in ll, 16-20 may be put thus: If long and short, broad and narrow, deep and shallow are the characteristics of the lines, the planes, the solids respectively, either (1) what is broad or narrow is not long or short, so that lines and planes are quite cut off from each other, or (2) it is, in which case the plane is a line.
19-20. The point of ém 8é. .. dmod00jcetar; seems to be: ‘ How will they describe the principles of excess and defect which are to angles and figures what the long and short is to the line, the broad and narrow to the plane, &c. ?’
20. tadtd te cupPaiver tots mept Tov dpiOudy, sc. cvuPaivovow, Long and short, &c., are, as much as straight and curved, attributes (to be exact, they are properties, which are xa atrd in the latter of the senses stated in Am. Post. i. 4. 73° 34-» 3) of the line, not its matter, just as great and small are attributes, not matter, of number, A. 992» 2, N. 10884 17.
23-31. Aristotle now introduces, in the middle of his discussion of the principles from which the Platonists derived geometrical objects, an argument against the general theory of ideal numbers existing apart from sensible things. The section breaks the continuity of the thought, and is evidently out of place.
23. wdvtwv ... ToUTwy appears to be neuter—‘all these cases’.
24. Tdv elddv Tdv ws yévous, cf. A. 991°31 Nn.
25. 97. Alexander and Bonitz take this in the sense of ‘ ascribes separate existence to’, but it seems doubtful if the word can mean this; nor is either of the suggested emendations very plausible. I believe that 07 has its ordinary meaning and that the argument runs thus : ‘A question which can be applied to all these Platonic beliefs is that which must be faced in the case of species of a genus, when one posits the existence of universals, viz. whether it is ‘animal’ itself that is present in the particular species of animal, or an ‘ animal’ distinct from ‘animal’ itself. If we do not assign separate existence to the
universal there is no difficulty; if we do assign separate existence to the One and the numbers, the question is difficult, not to say impossible.’
26. % Erepov atrod tuou. Jaeger reads Ceov for Gwov and renders ‘or an animal distinct from the sensible animal’. But this would be a mere synonym for rd €Gov avré, whereas mérepov ... 7 suggests alternatives. The meaning is made clear by Il. 29-31.
27-28. xwprotod 8¢... tod évds Kal Tav dprOuay, ‘the One and the numbers being separate from sensible things’.
gi. It seems quite possible, pace Bz., to retain atrd voet tT, in the sense of ‘is one thinking a thing-itself (or Idea)?” Cf. 1079” 9.
32. Tovadtys UAys, the various forms of great and small, 1. 9.
érepou S€, apparently Speusippus, cf. 1. 13 n.
33- KatadAys Uys, cf. 1. gn. For the construction ddAys vAns otas 70 wAnGos cf. De An. 424” 2, G. A. 766” 13,
35-4. Aristotle here reduces those who made the material principle of peyéy ‘something analogous to rAjOos’ (I. 33) to the same dilemma to which he reduced those who made the principle ’ “the kinds of great and small’ (# 9-20).
b5. ék Tod évos kat mdyQous, the view of Speusippus, cf. #9 n., Z. 1028 21 n.
6. Gmws 8° ody Aéyouct, ‘ but however they speak’. A€yovar is par- ticiple, cf. Zop. 128 33, De Sensu 444% 18, Pol. 1319» 37.
J. Tos €k Tod évds...dopicorov. Plato is probably referred to, as well as Xenocrates. For the evidence cf. Robin, pp. 641-654.
Q. 6 8 =oi ek Tod Evds Krd., |. 7.
II. at adtat, which is read by the vefus versio and apparently by Alexander, gives a better sense than the manuscript reading atra. Christ’s conjecture airaé is nearer to the manuscripts, but the crasis does not appear to be used by Aristotle. The passage suggests 10822 20, but the point is not the same. There the question was, How are the units combined into numbers? here it is, How are the formal and the material principle combined?
On pigs and Oécis cf. 1082420. «pao is a kind of pigis (Zop. 122>26)—the kind that belongs to fluids; though sometimes the words are used interchangeably (Po/. 126218). Cf. Al. De Mist. xili. 228. 7, 25 Br.
18-22, From the supposition that each unit is derived from the One and a part of plurality, three difficulties might seem to be deduced : (1) each of the parts of plurality must be indivisible, so that a divisible is composed of indivisibles (which is absurd),—unless we are to make the part of plurality itself a plurality and the unit divisible (which is equally absurd); (2) the elements will be not the One and plurality but the One and a part of plurality; (3) this plurality of indivisible parts which is supposed to be an element of number will be itselfa number. But there is a difficulty in this interpretation. In 1. 33 number is said to be composed of indivisible parts. That each of the parts of plurality should be indivisible (1. 18) is, then, in itself no difficulty. d8vatperov (1. 18) . . . dvarperqv (I. 19) is not a complete
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objection ; the following words cai. . . évés must go withit. The first objection then is that each of the parts of plurality will be indivisible, and thus the material principle will not be +AOos at all; and there are only two objections, not three. re, then, in 1 18 is caught up not by xaé in |. 19 but by éru in |. 21. :
In all this, Bz. remarks, Aristotle ignores the facts (1) that Plato probably did not think of number as containing units at all, (2) that he meant by the material principle of number something which was a mere potentiality and could not be charged with being an actual number. But as regards the first point, we must remember that Aristotle is here attacking not Plato but Speusippus (cf. 1. 5 n.), who did not believe in ideal but only in ordinary mathematical number (cf. 1076® 20-21 n.), and is therefore more open to Aristotle’s attack.
22. TS yap TAGs ddvaiperwv éotiv dprbuds, cf. Z. 1039 12 N.
23. Kat mepl Tods ofrw A€yovtas kTA. A similar question has already (1083> 36) been asked regarding the theory which derived number from the One and the indefinite dyad; Aristotle is now considering the theory which derives number from the One and rA7j@o0s. But it is not quite the same question that is asked. In 1083 36 the question was, Is number finite or infinite? Here the question is, Is the number which we have shown the original rA7#60s to be (1. 22) finite or infinite? That this is what Aristotle is asking is shown by |. 26 zotov ovv tAROos oTorxeiov.
tept tods otw Néyovtas. The manuscripts read apa xrA., and Bz, Ind. Ar. 562% 7 interprets wapa as ‘according to’, but this use seems unparalleled in Aristotle. Alexander seems not to have had the pre- position, and to have taken rots ovrw A€yovras as object of Cyryréov ; but this is not an Aristotelian construction.
24. xa, at first sight difficult, may be explained by the following para- phrase: ‘Since the units produced from the 7AjOos and the One were finite, the ThIGos also must have been finite ’.
26. €or. te . . . daeipov. Alexander explains ‘and plurality is different from infinite plurality’. (Though he does not use the word air, it would not be safe to infer with Christ that he did not read it.) So too Bz. The only objection to this interpretation is that if we adopt it, Il. 24-26 give no reason at all why the original plurality should not have been finite. But Aristotle evidently thinks that he is putting Speusippus into a difficulty. Perhaps, then, we should in- terpret the whole passage thus: ‘ There was, according to Speusippus, a finite plurality from which and the One the units were derived. And there is another plurality which is “ plurality itself’ and infinite plurality. Which sort of plurality, then, is the material principle in number ?’
27-34. Aristotle now passes from the views of Speusippus about the derivation of numbers to his views about the derivation of spatial magnitudes (stated in 432-34). The objections stated here to the
derivation of points are clearly akin to those raised against the deriva- tion of units in Il. 12-21. Line 29 answers to 14; 30, 31 to 15-173 31-34 (less closely) to 17-21.
29. ob yap... atry, ‘for this is not the one point which alone exists’,
36. tpdmous. Alexander knew both this reading and zpdrovs, and preferred the latter, interpreting it as rods évdoforépous (782. 25). But this is awkward, considering that Aristotle uses 6 rpéros immediately after (1086 11) in the chronological sense. It is equally awkward to suppose that Plato, Speusippus, and Xenocrates could be grouped - as oi mp@ro in the chronological sense, since in 1086411 Plato is distinguished from the other two as 6 zpéros. It seems better, then, to read with EJ zpdzovs, which is used quite similarly in 1080) 4, 10, 35, 1083 2, 1086231. There is perhaps, as Goebel remarks, a special appropriateness in the use of the musical term zpdzos with Siadwveiv.
1086? 2. oi pév, sc. Speusippus. Cf. 10762 20-21 n., 1080) 14,
5. ot 8¢, sc. Xenocrates. Cf. 1076% 20-21 n., 1080 22, 1083? 2.
5-6. 1a eidy .. . wovetv, ‘ wishing to make the Ideas at the same time also numbers ’.
7. Alexander (783. 3) seems to have omitted tds, and Jaeger follows him, taking ravtas as = 7a «ldy. But there would be no special point here in the description of the eidy as dpyad. A comparison with 10814 12-17, N. 1090” 36 shows the point to be that if the One and the great-and-small are taken as the first principles, it is hard to deduce do/h the Ideas and mathematical numbers from them. If tavtas be kept, the reference is to the discussion of the One and the great-and-small in 1083 23—1085 34. But ras airds would be rather more pointed, and may be the true reading.