^{page 247 note 1} Cp. Trendelenburg, Platonis de ideis et numeris doctrina ex Artitotele illustrate, p. 80 : ‘locos inter se, ut videtur, repugnantes.’ Zeller, Plat. Stud. p. 243: ‘mit welcher Stelle, die ihr wiedersprechende,’ etc.

^{page 247 note 2} Cp. Trend. I.e.: ‘his inter se collatis, alterum, ideas enm definiri numerum, qui habeat prius et posterius, altero, non ideas factas eorom in quibus sit prins et posterins, prorsus repugn at.’ Brandis, Ehein. Mus. 2 (1828), p. 563. Zeller, Phil. d. Gr., 3rd ed., II. i. p. 571: ‘Wie lässt sich nun aber mit dieser Auffassung…die Angabe vereinigen dass Plato und seine Schule von demjenigen in dem das Vor und Nach stattfindet, keine Ideen angenommen haben?’ Zeller himself, who at first followed Trendelenburg, but eventually realised the essentials of the interpretation of the Ethics passage, does not happen to say that the difficulty was merely due to this mistranslation, his attention being taken up with the questions about the meaning of raised by those who misunderstood the Ethics text.

^{page 249 note 1} This passage is considered below, § 4, paragraph 8.

^{page 250 note 1} About this principle there is no confusion within mathematics proper : indeed no occasion for it arises there. But it is otherwise with what may be called ‘reflective mathematics’ or ‘quasi-philosophic mathematics.’ For the attempt to find continuity within number itself (cf. Dedekind) is a mistake which comes from looking on the Numbers as magnitudes, and not realising the truth attained so long ago in Greek philosophy that they are Universals. A parallel mistake would be to treat ‘triangularity,’ ‘squareness,’ etc., etc., as figures, misled by the linguistic equivalence to them of ‘the Triangle,’ ‘the Square,’ etc.

^{page 251 note 2} As is well known, there are two criticisms in the Metaphysics of the Platonic theory of Ideas, one bk. A, and another in bk. M. The second is a kind of revised and expanded version of the first: in some places it is a mere duplicate, lengthy passages being repeated word for word. It is noteworthy th it whereas the version in bk. M has an introduction comes not in which may well imply the writer held it not strictly accurate to speak of the Idea-number theory as if due to Plato himself, as was done in the first version— (1078^a 9), later on in the same book the conception of is attributed to Plato. Cf. 1083^b 33, . This so far confirms the view which is now put forward. But the important point is the contention that the conception of the even if formulated by those who originated the Idea-number theory depends on the principles of the earlier theory alone.

The impression that the writer of the second version wishes to dissociate the name of Plato from the Idea-nnmber theory, is confirmed by another interesting circumstance. The second version while repeating the greater part of a long passage at the beginning of the first version nearly word for word, omits a little way down in the context that passage in the earlier version which involves the identification of Plato's Idea theory with the Idea-number theory and substitutes for it some different matter.

^{page 251 note 1} See below, § 8, on and in the Republic.

^{page 251 note 2} The writer finds that his friend Mr. J. A. Smith, of Balliol College, has independently arrived at much the same result; and also shares the opinion that the conception of was connected with Platonism in general, and not specially with the Idea-number theory.

^{page 252 note 1} undoubtedly included geometrical figures as well as number. Cp. Met. B. ii. 997^b 2 must include geometry. This seems also the clear implication of what is of geometry in 997^b 26 sqq. Compare also Metaph. A. ix. 991^b 27, quoted below,—, which that comprises more than .

It would not be necessary to mention this were not that Met. B. ii. 992^b 13 sqq. might cause difficulty: . Here it is implied that geometrical objects, in some sense, not included in , and so are not among . But Aristotle is not referring to the ordinary form of the doctrine of . He is attacking certain Platonists who, while identifying all with numbers, and retaining the view of the interin ediacy of the objects of mathematical science, put, may be supposed, the Universals of geometrical figures not among the Ideas (which were numbers), but degree below them (), thus introducing a fourth kind of object between the Ideas and The geometrical figures which were objects of geometry would still be found in the . For these Platonists see 1080^b 23, where Aristotle, after noticing certain differences among the Platonists about Ideal and Mathematical Number, says there were differences also about geometrical objects : , which shews that they had two kinds geometrical objects, one the mathematical and other . Cp. also 1028^b 25, where philosophers are distinguished from Plato and Speusippus: —

^{page 252 note 2} Omitting with Christ the .

^{page 252 note 3} The editors have a comma after and none after which gives what seems a wrong sense.

^{page 252 note 4} See above, § 3, paragraph 3.

^{page 253 note 1} See § 8,—On the expression .

^{page 253 note 2} So also Schwegler, note on Metaph. B iii. 16. See below note to paragraph 5.

^{page 253 note 3} Christ would emend either to . But the text is doubtless sound, corresponds to . That should be assigned to in another sense in the same context ought to cause no difficulty. Such carelessness is common in Aristotle, and there are far harsher instances to be found than this.

^{page 254 note 1} Hence if Metaph. 999^a 6 referred to the Platonist theory, the there spoken of could only be . But there is really no reference to Platonism in the passage : see below, § 7.

^{page 254 note 2} Zeller's utterances however (Phil. d. Gr. 3rd Edn. ii. 1. p. 570) are very confused. It is not necessary to discuss the nature of the confusion, as he has misunderstood the meaning of the and of the .

^{page 255 note 1} . The meaning of an objection brought in these words is that in the process of referred to the povdties would hare to be coordinate, whereas (exhyp.) as one must be necessarily prior to the other. Christ notices that Alexander thinks ,Preferable and himself conjectures , which can hardly be due to anything else than a misunderstanding. Bonitz appears to have taken the place rightly, but perhaps has not brought the point out clearly enough to prevent Christ's misconception.

^{page 255 note 2} The current expression ‘logical order,’ which we often use for some order not a time order, is vague. It seems properly to mean an order determined by some principle or conception, and thus should include every kind of numerical order, to which nevertheless it is sometimes opposed.

^{page 256 note 1} Endemns' attempt to explaiD this case in Eud. Eih. I. viii., where he reproduces Nic. Eth. I. vi. 1096^a 17 sqq., is an entire misunderstanding.

^{page 257 note 1} There is a curious slip here in the Teubner edition:—‘ codd. edd., emendavi.’