b4. We now add (3) that on this theory no body can be divided. For it would have to be divided at a plane, the plane at a line, the line at a point, so that since the point is on this view in- divisible, the body is so too; and if mathematical body, then also the sensible body in which it is.
11. (B) Nor can mathematical objects exist apart from sensible things. For (1) if there are separate mathematical solids, there will be (@) separate planes, lines, and points,
16. and therefore also besides (4) the planes, lines, and points of the mathematical solid there must be (c) planes, lines, and points prior to (while the former are simultaneous with) the mathematical solid.
24. Again there will be (d) lines and points prior to the lines in these planes, and (¢) points prior to those in these prior lines,
28. The accumulation is ridiculous; there is one set of solids apart from the sensibles, three sets of planes, four of lines, five of points. Which will be the objects of the mathematical sciences ?
36. (2) The same argument can be applied to numbers. There
Meera O70r20 —. 1076" 33 41
wil be units apart from the points, units apart from the objects of sense, units apart from the objects of knowledge.
39. (3) The objects of astronomy will exist apart from sensible things, as much as geometrical objects; but how can there be moving objects such as the heavens apart from sensible things?
1077* 4. There will be objects of optics and of harmonics—voice, senses, sensibles, animals, all separate from the ordinary objects of sense. ;
g. (4) There will be separate objects, which are neither numbers, points, spaces, nor times, for the universal mathematics which is true of all of these alike.
14. (5) The belief violates common sense. It makes mathematical objects prior to sensible things, but really they are posterior in sub- stance, being incomplete.
20. (6) What gives them unity? Things in ¢4zs world are made one by soul, by a portion of soul, or the like, but what gives unity to these divisible quanta?
24. (7) Length is generated first, then breadth, then depth, so that if that which is posterior in becoming is prior in substance, body is prior to the plane or line; and it is more complete because it is what becomes the vehicle of soul.
gi. (8) Body is a substance, but lines cannot be substances, either as form, or as matter (how could a thing be composed of lines ?). :
g6. They may be prior in definition to body, but they are not therefore prior in substance. That is prior in substance which excels in power of separate existence ; that is prior in definition whose defini- tion is implied in the definition of something else.
b4. If attributes cannot exist apart from substances, they are prior in definition to the complex of substance + attribute, but not in sub- stance ; thus the product of abstraction is not prior nor that of addition posterior.
12. Mathematical objects, then, are not more substantial than bodies, nor prior to them in being (but only in definition), nor separately existent; and since they could not be zz sensible objects either (1076 38-» 11), they exist either not at all or in some qualified sense.
1076? 33. # év Tots aioOntois etvar adtdé. This doctrine is said (1. 39) to have been discussed év rots dvaropyuacw, i.e. in Bk. B, and the reference is clearly to 998%7-19. The view in question is there described in a way which marks it off clearly from the ordinary
412 Commentary
Pythagorean view (for which see A. 987 27-29), and is attacked both there (998# 11-13) and here (» 1-3) by arguments which have force only against believers in separate Ideas of some kind. Further, in 10802 37—» 3 the regular Pythagorean view is. distinguished from another form of belief in numbers immanent in things (sc. the view referred to here), We may infer that Aristotle is speaking here either of Platonizing Pythagoreans (as Robin infers) or of Pythagoreanizing Platonists. For evidence of a Platonizing school of Pythagoreans cf. Robin, 649-651. Syrianus (84. 21) says that no Pythagoreans or Platonists held this view; but this is part of his general policy of defence of Platonism against Aristotle.
39. elpynror pev, B. 998% 11-15, 997% 12-34.
by, It is rather surprising that these thinkers recognized mathema- tical solids as distinct from sensible solids. Planes, lines, and points might naturally be distinguished from sensible things, since all sensible things have three dimensions; but what difference could there be be- tween mathematical and sensible sods? The answer no doubt is that by mathematical solids were meant the regular solids to which sensible objects never do more than approximate.
2. Tas dAas Suvdpers Kal pdcers, Alexander (725. 21) thinks the limits of sensible bodies, i. e. planes and lines, or else the characteristics studied by applied sciences like optics and harmonics, are meant. The meaning is fixed, however, by the corresponding passage in B. 998% 11-13. The dvvapes and dices are, quite generally, the characteristics of things, which these thinkers treated as separate Forms when consistency required that they, like ra poa@npuarixa, should be viewed as immanent in things.
4-11. Aristotle argues as follows: ‘If these mathematical solids are divisible, they are divisible along or at (karé) planes, and similarly the planes are divisible along lines, and the lines at points. But points are indivisible; so therefore are the lines, planes, and solids. But if the mathematical solids are indivisible, so must the sensible solids be.: Which is absurd.’ He treats the divisibility of the line at a point as implying the division of the point, and one might be disposed to question this. But the one does imply the other, according to the principles of the view he is criticizing, for (1) these thinkers cannot say, aS he would, that the point is brought into actual existence by the act of division; it is a substance, always existing actually ; and (2) they cannot say that the division comes de/ween two consecutive points, since (so Alexander says, and we may suppose that he is right) they, like Aristotle, held the line to be continuous and so to have no consecutive points.
1I—1077? 14. Aristotle now proceeds to the second alternative, stated in ® 34, that mathematical objects exist apart from sensibles (the view of Plato and of Speusippus). If there are mathematical solids apart from and logically prior to the sensible solids, there will be
(1) (Il. 14-16) planes, lines, and points apart from sensible planes, lines, and points.
(2) (Il. 16-24) planes, lines, and points in the mathematical solids.
(3) 4s planes, lines, and points apart from those numbered 2). (4) (il. 24-27) lines and points prior to the lines in the planes numbered (3).
(5) (ll. 27, 28) points prior to those in the lines numbered (4).
The lines in the planes numbered (3), though introduced by raw (1. 24) are not meant to be a new class not mentioned before ; else we should get five classes of lines instead of (as Aristotle says, 1. 32) four. They are identified with the lines numbered (3), and are mentioned anew in ]. 25 only as leading up to the further class of lines and points numbered (4).
The enumeration is careless and by no means complete. Aristotle might have argued that if there are two sets of solids (sensible and mathematical), there will be two sets of planes in these solids and two sets abstracted from them; four sets of lines in these planes and four sets abstracted from them ; eight sets of points in these lines and eight abstracted from them.
The enumeration betrays its incompleteness by lack of symmetry. If we take the sets of mathematical objects which he mentions we get the series:—one set of solids, three of planes, four of lines, five of points; and if we add in the sensibles we get the series 2, 4, 5, 6. Neither series is symmetrical. The fact is that Aristotle begins cor- rectly the geometrical series 2, 4, 8, 16, but tires of the capevors and turnstheseries intoan arithmetical one. The capevors is éroros enough, even as stated by him. The error which lies at its base is the ywp- opos Of, or assigning of separate existence to, what is only distinguish- able by thought.
QI. akwhytos = pabyparcKors.
38. For ra dvra aicOnra of the manuscripts it seems necessary to read Ta Ova, Ta aicOynTd.
39. ev Tots dwophpacw, B. 997) 12-34.
1077 2. Bz.’s éorat is confirmed by |. 5 and B. 997? 16.
3. otpavey, sc. rapa Tov aicOyrov ovpavor, cf. B. 997° 16.
Q. ypdperar, Alexander explains, means detkvutar. Cf. Top. 158” 30 od padiws ypadeo Oar and the use of didéypaypa = proposition in Cas. 14° 39, B. 998% 25, A. ro14236. In all these cases it would seem that a proof aided by a figure is meant. The general mathematics here referred to, which proves attributes that are not peculiar to numbers or to spatial magnitudes or to times, is also mentioned in 1077) 14, E. 1026827, An. Post, 74223. Eudoxus’ doctrine of proportion, which is preserved in Euclid’s Lvemen/s, Bk. V, is the best instance of this ‘ general mathematics’ (cf. 74 23).
20-24. et... . cuppever is a digression from the main point. The question what causes the unity of mathematical magnitudes is inserted between two arguments from genesis, (1) the argument that sensible magnitudes must be prior in essence to mathematical because they
414 Commentary
are later in generation (Il. 18-20), and (2) the argument that solids must be prior in essence to planes and lines because they are later in generation (Il. 24-28).
20. Bz.’s ri Kat mor’, ‘by virtue of what in the world’ is attractive, but though ris wore is common ris cai wore does not seem to be recorded | as occurring in this sense, and itis better to keep Bekker’s reading tive KOL ToT, -
22, wépet uxis, e.2., Says aden? in the case of animals which have only the sense of touch and therefore only a part of the soul.
eUNoyov (sc. éorw ev etva) is Jaeger’s probable. emendation of cidoyo.
We should perhaps read ddAw tii, edAdyws. For etddyws added thus at the end of a sentence cf. Bz. Index 297% 22-27, Alexander illustrates dAAw tit edAdyw by the case of things glued or tied to- gether.
24-30. Bz. points out that there is a serious ambiguity in Aristotle’s use of yéveous in this argument. yéveous in the sense in which 76 yevéret vorepor is oaia mpdrepor is natural genesis, e.g. the growth of the boy into the man. But yéveous in the sense in which it can be applied to mathematical objects refers to the quite different process by which the line is generated by a moving point, the plane by a moving line, the solid by a moving plane. The ambiguity deprives the argument of whatever value it might otherwise have possessed.
24-26. mpdtov ... éoxev evidently refers to Speusippus fr. 4. 44-47 (Lang).
The two senses of kar’ otciay (or pvcer) tpdtepov answer to two of the meanings of otaéa which are so often distinguished by Aristotle. The first sense answers to that sense of ota/a in which it means form,
or to the rdde zu considered as a fully formed or developed thing ; the second to that in which it means 70 troxeiwevoy or the rdde te Con- sidered as something capable of separate existence.
BI. 78 yap exer mws Td Tedevoy, cf. De Caclo 2682 7-24. ws, says Alexander, because gwa mere mathematical solid it lacks the qualities by which 7a duoixa eidororetrar (732. 5):
b 3. Sowy ot Adyor ek TOv Adyov is difficult. The natural translation would be ‘those things are prior in definition whose definitions are com- pounded out of the definitions of the other things ’, but this is the exact opposite of Aristotle’s doctrine (cf. A. 1018? 34, Z. 1035» 4). We might interpret dcwy as depending on Adywy, not on Adyou, and translate ‘of whose definitions the definitions of the other things are com- pounded’, but it seems more likely that the transition from doa to dour leads Aristotle to substitute in thought an antecedent rovrwy for the antecedent raira. ‘Those things are prior in substance which when separated from other things surpass them in power of indepen- dent existence, and things are prior in definition to the things whose definitions are compounded out of their definitions.’ | Schwegler’s proposal to excise é« does not, therefore, seem necessary. For a similar confusion cf. Z. 1034? 31.
4. 00x dpa Smdépxer does not mean that these characteristics are never found together, but that they are not always found together. It is not necessary to read tdpyxer (de/) as has been suggested.
15. évedexeto, as was shown in 1076 38-} rr.