36-37. ‘ And so they identify the odd with the One (a principle, not a number); for if oddness had depended on the first odd numéer, how would 5 (which according to the theory has not the ideal 3 in it) be odd?’ To say that the number 3 is the principle of oddness would imply deriving 5 from the union of 2 and 3, whereas according to the Platonic view itis otherwise derived.
The force of 80 seems to be this: The Platonists derived all derivative entities either from the first principles or from the numbers up to 10. Now oddness could not be derived from a number such as 3, because this would not explain the oddness of any other number (the numbers being supposed independent of each other) ; sherefore it had to be derived from one of the principles, and of the two the One rather than the indefinite dyad was indicated for the purpose.
For the force of év rq rpiads cf. Eucken, Sprachgebrauch d. Ar. pie2as
37-b 2. éru... SexdSos, ‘ Further, magnitudes and the like extend, they say, only up to a certain point, e.g. there is the first or indivi- sible line, then the two, &c.; these entities also extend only up to ro.’ Aristotle is giving a further ground for his statement that the Pla- tonists treat 10 as the perfect or complete number (I. 31). They recognize first the primary or indivisible line (their substitute for the ‘point’ of geometrical theory, A. 992% 22). mpdérn ypappi) dropos is difficult, and it seems best to read 4} rpérn ypayyn, 4 dropos. (Alterna- tively we might omit % after otov as Schwegler proposes, and translate ‘first comes the indivisible line’; but 7 is more likely to have been omitted after zpary than to have been inserted after ofov.) I know of no exact parallel to zpwry ypaypy in this sense, but in A. 9924 21 the indivisible line is called dpy% ypappys, and 4 te dpxi mpGrov Kal 70 mpa@tov dpyyn, Top, 121? 9.
Certain Platonists (A. 992°21, De An. 404» 16-24 suggest that Plato himself was among them, while a comparison with N. 1090 20- 32 suggests that Xenocrates also is referred to) connected the point or indivisible line with the number 1, the line with 2, the plane with 3, the solid with 4 (N. rogob22, Z. 1036514, H. 1043%33); and I+24+3+4= Io.
b 4-13. The discussion whether the One or number is prior may be compared with the discussion in Z. 10, 11, where the same illustration (the right and acute angles) is used (1034> 28, 10356, 1036% 14). The present passage seems to be written without any reference to the previous one.
7. 6. dprotar Kat TO Adyw. Two reasons for the priority of the right angle are given, (1) that it is definite, while the acute angle may be of any size between 0° and 90°, and (2) that it is involved in the
Gea
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definition of the acute angle while the acute angle is not involved in zs definition.
12, 76 dpe, i.e. what is elsewhere called 76 é& doty (A. 1071" 9), or 76 ovvdpdw (H. 1043%22). 6 dpiOuds, which is here treated as a compound of form and matter, was in 1. 6 described as form.
15-16. ddXG tpdtrov Gov kth, An opposition of ‘indivisible in Aéyos’ and ‘indivisible in time’ would be quite unparalleled in Aristotle, and no reasonable meaning can be attached to it. Alexander explains that the universal is indivisible in Adyos because ‘ footed two-footed animal’ is not divisible into other Adyou and eidy as ‘animal’ is into ‘man’ and ‘horse ’—which is evidently nonsense ; and that the particular is indivisible in time because my form is not prior in time to me— which, besides interpreting 76 émi pépovs in a sense which we have seen reason to doubt, is a very unnatural interpretation of ‘indivisible in time’. Nor does any better interpretation of this last phrase seem possible. We are driven, then, to suppose that rpomov dAXov KTAi does not qualify ‘indivisible’. Ifwith Bekker we read a full stop before dAAd, we may take tpdrov addAov as qualifying dpyy. ‘The One is
said to be dpxy because it is indivisible. But the universal as well as the element is indivisible. Yes, but their consequent primariness is of different kinds.’ We thus get the ordinary opposition of zpdérepov Aoyw and xpovw, for which cf. ll. 12, 13, Z. 1028%32, 1038 27, @. 1049 11, Phys. 265% 22. Accordingly in]. 16 Aristotle asks not ‘in which sense is the One indivisible?’ but ‘in which sense is the One apxn Le
18. Kat ckatépa, pia may mean (1), as Alexander says, ‘ and each of these is one and the same with itself’. I.e. these words may simply emphasize the fact that one single thing may be in one sense prior, in another posterior, to another single thing. But it seems more likely (2) that these words fit into the argument as follows: In which sense is the One dpxyn? The right angle is prior to the acute angle, and in another sense the acute angle is prior to the right angle, and each of these is one (s¢.the acute angle is one as an element is one, and the right angle is one as a whole—here confused with a universal—is one). |The Platonists accordingly (84) make the One primary in both senses, in time and in Aéyos. But this is impossible ; for ‘One’ is here used ambiguously— it is what is one, as a form or essence (= universal) is one, that is primary xara Adyov, but it is what is one as a part, or as matter, that is primary kata xpovov.
20-24. éott ydp mws ktd. ‘For each of the two (the unit and the number) is one in a certain sense—in /rwth, if the number is not a mere aggregate but a unity consisting of units qualitatively distinct from those of any other number’ (cf. 10808 15-35), ‘each of its two units exists’ (and is one) ‘only potentially, not actually’ (while the number exists, and is one, actually). ‘This isthe truth, and the cause of the error into which the Platonists fell is that’ &c.
22. For owpds in this sense cf. Z. 1040? 9, 1041 12, H. 1044% 4, 1045° 9.
25. €k Tov Adywy Tov Ka8ddov. The Platonic method of inquiry is described similarly in A. 98731, ©. 105035, A. 10697 28, N. 1087) ar,
26. 16 év kal thy dpxyv, ‘the One, that is, the first principle’. The Platonists, Aristotle means, treated the One which was the formal principle of number as being at the same time to number what the point is to the line, viz. a material principle.
4 yap povas ottypy AWetds eotw. Cf. the definition of the point as povas Gow éxovoa De An. 40926, A. 1016” 25 n.
27. €tepot ties, the Atomists.
2g. It is evidently not the unit’s turning out to be matter and its turning out to be prior to the two that are meant to be described as dma, but its turning cut to be prior and its turning out to be posterior to the two, so thata commais wanted after dpu0yav. Cf. Robin, p. 396, who, however, puts a full stop after dpub.av.
30-32. 51d SE... ENeyov. ‘ But owing to the universal nature of their inquiries they treated the unity which can be predicated of every number as being in this way too a part’, sc. as a formal part
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or element in the definition predicable of each number, as well as a material part of it. For this line of thought about 76 év, which led the Platonists to treat it as the very essence of real things, cf. B. 996 4, 998 17, 10017 4, 20, I. 10539, 20, K. 1059 27, 1060? 3.
32. Taira 8°... bmdpyewv, ‘but one thing cannot be at the same time a material and a formal element in one other thing’. Cf. 1. 19.
32-34. ei S¢... apy. ‘ But if the One itself must only be without position (for it differs in no respect except in that it is a first principle),’ What the One differs from only by being a first principle is the wef; but its being without position distinguishes it from the pozn (I. 26). Thus the two clauses have not any such connexion as ydp indicates, and can hardly be right as they stand. Alexander feels no difficulty, but gives an impossible interpretation; and Bz’.s interpretation, which takes aerov as if it could mean dpyixdy, does nothing to meet the difficulty. The proposed emendations of dOerov (ddvaiperov Schwegler, dovvOerov Bywater) would give a satisfactory sense if it were not for pdvov, but in the presence of povov are unsatisfactory. Two suggestions may be made. (tr) It is just possible that a@eroy may be used in a new sense. Each unit has, on the Platonic view, a setting or Oéc1s in some parti- cular number ; all that distinguishes the One which is the formal prin- ciple of number is that it has no such particular setting, that it is aerov. It would not be unlike Aristotle to use dOeros thus in a different sense from that which it bore in l. 27, but the suggested use of d@eros is apparently without parallel. (2) We might suppose povaducoy to have been corrupted into pdvoy ddiov, and this to have been altered through a reminiscence of ]. 27 into povoy aberov.
33-34. The use of od0evi.. . 4 for otfevi dAdw... 7 OF OVMeL... aXX’ 7 is irregular, but cf. Kihner, ii. 2. §540, Anm. 4.
1085* 1-2. The argument is not, as Bz. says, the same as that in 1081 25-35. It is simply this: If one thing added to another always makes a two, the two itself and the three itself make a two, and a two of whose generation the Platonists can give no account. They cannot derive it from the One and the indefinite dyad, since what these originate is the two itself, the three itself, and so on.
3. ab pev odk oti, cf. 10822 20, Phys. 227220. Only those things touch one another ay ra dxpa dua 226 23, so that things which have not extent, such as units or numbers, cannot touch, though they can be successive. év rots dpufmots is ambiguous ; Aristotle means that there is not contact but only succession, both as between units in a number (1. 4) and as between numbers (I. 6).
4-5. oowy .. . tpidd. might be taken either with what precedes or with what follows. ‘The meaning may be (1) 1o 8 éeéqs éorw ev Tals povacw dowv pH eore petagd, Or (2) worepov at povddes, dowv py ore petagv, epeéjs. In either case 7 7H Tpiad. is embarrassing, since it is only the units in 2 that Aristotle goes on to speak of; but it is specially embarrassing if dowy «rd. be taken as in (2). Therefore (1) seems preferable. Then ai év 77 dudé. is to be understood as the subject of epeéqs (ior) in |. 5.
M. 8. 1084> 32 —g. 1085414 455
6. Bz. reads r@ ebe&fs and claims the authority of Alexander. But it is not clear what Alexander read, so that it seems better to retain the manuscript reading trav édeéjs, which gives a good sense. Aristotle does not here state the objections which follow if (1) the units, or one of the units, in two, or (2) two itself, are to succeed the number one directly. The objection to (1) is that then there is a two (composed of the number one + one of the units in two) before there is the number two itself (cf. 10818 32). The objection to (2) is that the first unit in two, being prior to the second unit, should be prior to the two composed of them (10818 25~27),
7. tav botepov yevav Tod dpOyod, ‘the kinds posterior to number’. These are also called ra wera tods dpiOuovs A. 99213, ra pera Tas ideas M. 1080» 25, For the priority of numbers to geometrical objects cf, A. 982226, Z. 1028" 21 n.
Q. ot pev ydp. érepor d¢ does not come till 1. 32. The Platonic opinions mentioned regarding the material principle of spatial magni- tudes are
(1) That it is the various kinds of the great and the small (108529, A. 992%11, N. 1090537). Some, if we may believe Aristotle, did not distinguish the great and the small which is the material principle of number from that which is the material principle of spatial magnitudes (B. roo1» 19). Al. 228. 10, Asc. 207. 37, Syr. 48. 20 think Plato himself is here referred to, and this may well be so. Others divided the great and small into the many and few, which is the épy7 of number, and the long and short, the broad and narrow, the deep and shallow, which are the dpxaé of spatial magnitudes (N. 1089? 11).