[197] Plato, Republic, vi. p. 508 E. Τοῦτο τοίνυν τὸ τὴν ἀλήθειαν παρέχον τοῖς γιγνωσκομένοις καὶ τῷ γιγνώσκοντι τὴν δύναμιν ἀποδιδὸν τὴν τοῦ ἀγαθοῦ ἰδέαν φάθι εἶναι, αἰτίαν δ’ ἐπιστήμης οὖσαν καὶ ἀληθείας ὡς γιγνωσκομένης, &c.
[198] Plato, Republic, vi. p. 509 B. Καὶ τοῖς γιγνωσκομένοις τοίνυν μὴ μόνον τὸ γιγνώσκεσθαι φάναι ὑπὸ τοῦ ἀγαθοῦ παρεῖναι, ἀλλὰ καὶ τὸ εἶναι τε καὶ τὴν οὐσίαν ὑπ’ ἐκείνου αὐτοις προσεῖναι, οὐκ οὐσίας ὄντος τοῦ ἀγαθοῦ, ἀλλ’ ἔτι ἐπέκεινα τῆς οὐσίας πρεσβείᾳ καὶ δυνάμει ὑπερέχοντος. Καὶ ὁ Γλαύκων μάλα γελοίως, Ἄπολλον, ἔφη, δαιμονίας ὑπερβολῆς! Σὺ γάρ, ἦν δ’ ἐγώ, αἴτιος, ἀναγκάζων τὰ ἐμοὶ δοκοῦντα περὶ αὐτοῦ λέγειν. — Also p. 509 A.
The Idea of Good rules the ideal or intelligible world, as the Sun rules the sensible or visible world.
Here then we have two distinct regions or genera; one, the conceivable or intelligible, ruled by the Idea of Good — the other the visible, ruled by the Sun, which is the offspring of Good. Now let us subdivide each of these regions or genera, into two portions. The two portions of the visible will be — first, real objects, visible such as animals, plants, works of art, &c. — second, the images or representations of these, such as shadows, reflexions in water or in mirrors, &c. The first of these two subdivisions will be greatly superior in clearness to the second: it will be distinguished from the second as truth is distinguished from not-truth.[199] Matter of knowledge is in the same relation to matter of opinion, as an original to its copy. Next, the conceivable or intelligible region must be subdivided into two portions, similarly related one to the other: the first of these portions will be analogous to the real objects of vision, the second to the images or representations of these objects: the first will thus be the Forms, Ideas, or Realities of Conception or Intellect — the second will be particular images or embodiments thereof.[200]
[199] Plato, Republic, vi. pp. 509-510. 510 A: διῃρῆσθαι ἀληθείᾳ τε καὶ μή, ὡς τὸ δοξαστὸν πρὸς τὸ γνωστόν, οὔτω τὸ ὁμοιωθὲν πρὸς τὸ ᾧ ὡμοιώθη.
[200] Plato, Republic, vi. p. 510 B.
To the intelligible world there are applicable two distinct modes of procedure — the Geometrical — the Dialectic. Geometrical procedure assumes diagrams.
Now in regard to these two portions of the conceivable or intelligible region, two different procedures of the mind are employed: the pure Dialectic, and the Geometrical, procedure. The Geometer or the Arithmetician begins with certain visible images, lines, figures, or numbered objects, of sense: he takes his departure from certain hypotheses or assumptions, such as given numbers, odd and even — given figures and angles, of three different sorts.[201] He assumes these as data without rendering account of them, or allowing them to be called in question, as if they were self-evident to every one. From these premisses he deduces his conclusions, carrying them down by uncontradicted steps to the solution of the problem which he is examining.[202] But though he has before his eyes the visible parallelogram inscribed on the sand, with its visible diagonal, and though all his propositions are affirmed respecting these — yet what he has really in his mind is something quite different — the Parallelogram per se, or the Form of a Parallelogram — the Form of a Diagonal, &c. The visible figure before him is used only as an image or representative of this self-existent form; which last he can contemplate only in conception, though all his propositions are intended to apply to it.[203] He is unable to take his departure directly from this Form, as from a first principle: he is forced to assume the visible figure as his point of departure, and cannot ascend above it: he treats it as something privileged and self-evident.[204]
[201] Plato, Republic, vi. p. 510 B. ᾗ το μὲν αὐτοῦ (τμῆμα) τοῖς τότε τμηθεῖσιν ὡς εἰκόσι χρωμένη (this is farther illustrated by p. 511 A — εἰκόσι χρωμένην αὐτοῖς τοῖς ὑπὸ τῶν κάτω ἀπεικασθεῖσἰ) ψυχὴ ζητεῖν ἀναγκάζεται ἐξ ὑποθέσεων, οὐκ ἐπ’ ἀρχὴν πορευομένη ἀλλ’ ἐπὶ τελευτήν, &c.
[202] Plato, Republic, vi. p. 510 C-D. οἱ περὶ τὰς γεωμετρίας τε καὶ λογισμοὺς καὶ τὰ τοιαῦτα πραγματευόμενοι, ὑποθέμενοι τό τε περιττὸν καὶ τὸ ἄρτιον καὶ τὰ σχήματα καὶ γωνιῶν τριττὰ εἴδη καὶ ἄλλα τούτων ἀδελφὰ καθ’ ἑκάστην μέθοδον, ταῦτα μὲν ὡς εἰδότες, ποιησάμενοι ὑποθέσεις αὐτά, οὐδένα λόγον οὔτε αὑτοῖς οὔτε τοῖς ἄλλοις ἔτι ἀξιοῦσι περὶ αὐτῶν διδόναι, ὡς παντὶ φανερῶν· ἐκ τούτων δ’ ἀρχόμενοι τὰ λοιπὰ ἤδη διεξιόντες τελευτῶσιν ὁμολογουμένως ἐπὶ τοῦτο, οὖ ἂν ἐπὶ σκέψιν ὁρμήσωσιν.