[32]See chap. 33. [33]These lines are as the numbers 4, 3, 2. For 4 to 3 is sesquitertian, 3 to 2 is sesquialter, and 2 is an arithmetical medium between 4 and 3. [34]For an explanation of this assertion of Plato in the Republic, see my Theoretic Arithmetic. [35]“The Pythagoreans,” (says Syrianus in Aristot. Metaphys. lib. 13.) “received from the theology of Orpheus, the principles of intelligible and intellectual numbers, they assigned them an abundant progression, and extended their dominion as far as to sensibles themselves.” Hence that proverb was peculiar to the Pythagoreans, that _all things are assimilated to number_. Pythagoras, therefore, in the Sacred Discourse, clearly says, that “number is the ruler of forms and ideas, and is the cause of Gods and dæmons.” He also supposes, that “to the most ancient and artificially ruling deity, number is the canon, the artificial reason, the intellect also, and the most undeviating balance of the composition and generation of all things.” αυτος μεν Πυθαγορας, εν τῳ ιερῳ λογῳ, διαρρηδην μορφων και ιδεων κραντορα τον αριθμον ελεγεν ειναι, και θεων και δαιμονων αιτιον· και τῳ πρεσβυτατῳ και κρατιστευοντι τεχνιτῃ θεῳ κανονα, και λογον τεχνικον, νουν τε και σταθμαν ακλινεσταταν τον αριθμον υπεικε συστασιος και γενεσεως των παντων. Syrianus adds, “But Philolaus declared that number is the governing and self-begotten bond of the eternal permanency of mundane natures.” Φιλολαυς δε, της των κοσμικων αιωνιας διαμονης την κρατιστευουσαν και αυτογενη συοχην ειναι απεφῃνατο τον αριθμον. “And Hippasus, and all those who were destined to a quinquennial silence, called number the judicial instrument of the maker of the universe, and the first paradigm of mundane fabrication.” οι δε περι Ιππασον ακουσματικοι ειπον κριτικον κοσμουργου θεου οργανον, και παραδειγμα πρωτον κοσμοποιϊας. “But how is it possible they could have spoken thus sublimely of number, unless they had considered it as possessing an essence separate from sensible, and a transcendency fabricative, and at the same time paradigmatic?”