James Adam, The Republic of Plato

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ταύτης in spite of γεωμετρίαις because Geometry is itself one art: cf. VII 533 C γεωμετρίας τε καὶ τὰς ταύτῃ ἑπομένας. The plural γεωμετρίαις does not mean the ‘various branches of geometry’ (as D. and V. suppose), but geometrical investigations: cf. λογισμούς for ‘Arithmetic’ in 510 C.

αὐτὸς ὁ λόγος κτλ.: ‘the argument grasps by itself, through the power of dialectic.’ λόγος is not the faculty of reason (“Vernunft” Schleiermacher), which is νοῦς, or even ‘thought’ (“Gedanke” Schneider), but rather “the impersonal reason, or drift of the argument” (Bosanquet), the instrument by which νοῦς works (Krohn Pl. St. p. 140). ὁ λόγος is of course personified, as it constantly is in this sense.

δυνάμει should not be translated ‘faculty,’ but simply ‘power’ (cf. 508 E note): the argument, unaided by εἰκόνες (αὐτός ‘by itself,’ cf. αὐτοῖς εἴδεσι 510 B note), grasps its object by the inherent power of dialectical argumentation (διαλέγεσθαι), and nothing else. In spite of Grimmelt (de reip. unit. etc. p. 52) it is certainly an error to identify ὁ λόγος with νοῦς. Why does Dialectic dispense with all sensible images or illustrations? Plato (it should be remembered) holds that the intrusion of any element of sense-perception, however small, impedes the exercise of thought: see Phaed. 79 C ff. The ὑποθέσεις of the dialectician may be and often are generalisations from αἰσθητά, but a generalisation, regarded in itself, is wholly νοητόν. These ὑποθέσεις it is the province of Dialectic to test in every possible way, to demolish where necessary (VII 533 C note), to correct by one another, to classify according to their mutual coherence and interdependence, until by an exhaustive scrutiny of all νοητά we grasp the unifying principle of all existence—the Idea of the Good. Cf. VII 517 C note and see on τοῦ ἀνυποθέτου below and the Appendix to Book VII On Plato's Dialectic.

τῷ ὄντι indicates that we are to take the word in its literal etymological signification, ‘literally hypotheses or underpositions, stepping-stones as it were and starting-points.’ For this use of τῷ ὄντι and kindred expressions see I 343 C, V 474 A notes and W. G. Headlam On editing Aeschylus pp. 138 ff. With ἐπιβάσεις cf. Symp. 211 C ὥσπερ ἐπαναβαθμοῖς χρώμενον.

τοῦ ἀνυποθέτου. See on ἀρχὴν ἀνυπόθετον 510 B. Plato makes no attempt in the Republic to classify Ideas in such an ascending scale as he here suggests, though it is probable from 509 A that Knowledge and Truth would rank near to the Good. Nor is there any dialogue in which an exhaustive classification is even attempted. Such hints as Plato gives us throughout his writings are enumerated in Stumpf das Verhältniss etc. pp. 50, 56, 76, and in Zeller^{4} II 1, pp. 704—707: cf. also Fouillée La Philosophie de Platon II pp. 99—104. We must suppose that each higher Idea will excel all the lower both in range and in excellence. These two characteristics are, from Plato's point of view, the same. The wider an Idea is in range and extension, the greater will be the sum of existences of which it is the cause. But the Idea of Good is the cause of all existence, so that each higher Idea will be better than all below it, because it contains more of Good. Beyond this it is perhaps safer not to go. A systematic attempt to correlate all intelligibles among themselves and in their connexion with the Good would have been premature in Plato's day, and is premature still. The permanent value of Plato's conception lies in the ideal which it sets before every succeeding generation of investigators.

πάλιν αὖ κτλ. The dialectician's progress involves both an ascent and a descent—an ascent ἐπὶ τὴν ἀρχήν, and a descent ἀπὸ τῆς ἀρχῆς ἐπὶ τὴν τελευτήν (cf. Aristotle quoted on 510 B). By the time that he reaches the Idea of the Good, all his surviving ὑποθέσεις have become exact counterparts of the Ideas which are their objective correlates; the others have all of them been demolished (VII 533 C note). The conclusions (τελευταί) of dialectic are therefore impregnable; ψευδὴς ἐπιστήμη is a contradiction in terms (V 477 E note). For more on this subject see the Appendix to Book VII On Plato's Dialectic.