Chapter XVIII.—Pythagoras’ System of Numbers.

Pythagoras, then, declared the originating principle of the universe to be the unbegotten monad, and the generated duad, and the rest of the numbers. And he says that the monad is the father of the duad, and the duad the mother of all things that are being begotten—the begotten one (being mother) of the things that are begotten. And Zaratas, the pupil of Pythagoras, was in the habit of denominating unity a father, and duality a mother. For the duad has been generated from the monad, according to Pythagoras; and the monad is male and primary, but the duad female (and secondary). And from the duad, again, as Pythagoras states, (are generated) the triad and the succeeding numbers up to ten. For Pythagoras is aware that this is the only perfect number—I mean the decade—for that eleven and twelve are an addition and repetition of the decade; not, however, that what is added634    Miller would read for προστιθέμενον, νομιστέον or νομίζει. constitutes the generation of another number. And all solid bodies he generates from incorporeal (essences). For he asserts that an element and principle of both corporeal and incorporeal entities is the point which is indivisible. And from a point, he says, is generated a line, and from a line a surface; and a surface flowing out into a height becomes, he says, a solid body. Whence also the Pythagoreans have a certain object of adjuration, viz., the concord of the four elements. And they swear in these words:—

“By him who to our head quaternion gives,

A font that has the roots of everlasting nature.”635    Respecting these lines, Miller refers us to Fabricius, in Sextum Empiricum, p. 332.

Now the quaternion is the originating principle of natural and solid bodies, as the monad of intelligible ones. And that likewise the quaternion generates,636    The Abbe Cruice adduces a passage from Suidas (on the word ἀριθμός) which contains a similar statement to that furnished by Hippolytus. he says, the perfect number, as in the case of intelligibles (the monad) does the decade, they teach thus. If any, beginning to number, says one, and adds two, then in like manner three, these (together) will be six, and to these (add) moreover four, the entire (sum), in like manner, will be ten. For one, two, three, four, become ten, the perfect number. Thus, he says, the quaternion in every respect imitated the intelligible monad, which was able to generate a perfect number.

[23] Πυθαγόρας τοίνυν ἀρχὴν τῶν ὅλων ἀγέννητον ἀπεφήνατο τὴν μονάδα, γεννητὴν δὲ τὴν δυάδα καὶ πάντας τοὺς ἄλλους ἀριθμούς. καὶ τῆς μὲν δυάδος πατέρα φησὶν εἶναι τὴν μονάδα, πάντων δὲ τῶν γεννωμένων μητέρα [τὴν] δυάδα, γεννητὴν γεννητῶν: καὶ [γὰρ] Ζαράτας ὁ Πυθαγόρου διδάσκαλος ἐκάλει τὸ μὲν ἓν πατέρα, τὰ δὲ δύο μητέρα. (γ)εγέννηται γοῦν ἐκ μὲν μονάδος δυὰς κατὰ τὸν Πυθαγόραν, καὶ ἔστιν ἡ μὲν μονὰς ἄρρεν καὶ πρώτη, ἡ δὲ δυὰς θῆλυ [καὶ δευτέρα]. παρὰ τῆς δυάδος δὲ πάλιν, ὡς ὁ Πυθαγόρας λέγει, [γίνεται] ἡ τριὰς καὶ οἱ ἐφεξῆς ἀριθμοὶ μέχρι τῶν δέκα. τοῦτον γὰρ οἶδε μόνον τέλειον ἀριθμὸν [ὁ] Πυθαγόρας, τὸν δέκα: τὸν γὰρ ἕνδεκα καὶ δώδεκα προσθήκην καὶ ἐπαναποδισμὸν τῆς δεκάδος, οὐκ ἄλλου τινὸς ἀριθμοῦ γένεσιν [εἶναι] [τὸ προστιθέμενον]. πάντα δὲ σώματα στερεὰ ἐξ ἀσωμάτων γεννᾷ: τῶν τε γὰρ σωμάτων καὶ ἀσωμάτων ὁμοῦ στοιχεῖον εἶναί φησι καὶ ἀρχὴν τὸ σημεῖον, ὅ ἐστιν ἀμερές. γίνετ(αι) δέ, φησίν, ἐκ σημείου γραμμὴ καὶ [ἐκ γραμμῆς ἐπιφάνεια,] ἐπιφάνεια δὲ ῥυεῖσα εἰς βάθος στερεὸν ὑφέστηκε, φησί, σῶμα. ὅθεν καὶ ὅρκος τίς ἐστι τοῖς Πυθαγορικοῖς ἡ τῶν τεσσάρων στοιχείων συμφωνία: ὀμνύουσι δ' οὕτως: ναὶ μὰ τὸν ἁμετέρᾳ κεφαλᾷ παραδόντα τετρακτύν, πηγὴν ἀενάου φύσεως ῥι(ζώμ)ατ' ἔχουσαν. ἔστι γὰρ ἡ τετρακτὺς τῶν φυσικῶν καὶ στερεῶν σωμάτων ἀρχή, ὡς ἡ μονὰς τῶν νοητῶν. ὅτι δὲ καὶ ἡ τετρακτὺς γεννᾷ, φησί, τὸν τέλειον ἀριθμόν, ὡς ἐν τοῖς νοητοῖς [ἡ μονάς], τὸν δέκα, διδάσκουσιν οὕτως: εἰ ἀρξάμενος ἀριθμεῖν λέγει τις ὅτι ἕν, καὶ ἐπιφέρει δύο, ἔπειτα ὁμοίως τρία, ἔσονται ταῦτα ἕξ: πρὸς δὲ τούτοις ἔτι τέσσαρα, ἔσται [ὁμοίως] τὸ πᾶν δέκα: τὸ γὰρ ἕν, δύο, τρία, τέσσαρα γίνετα(ι) δέκα, ὁ τέλειος ἀριθμός. οὕτως, φησί, κατὰ πάντα (ἐ)μιμήσατο ἡ τετρακτὺς τὴν νοητὴν μονάδα, τέλειον ἀρι(θ)μὸν γεννῆσαι δυνηθεῖσαν.