Again, to start afresh, since of quantity one kind is viewed by itself, having no relation to anything else, as “even,” “odd,” “perfect,” and the like, and the other is relative to something else and is conceived of together with its relationship to another thing, like “double,” “greater,” “smaller,” “half,” “one and one-half times,” “one and one-third times,” and so forth, it is clear that two scientific methods will lay hold of and deal with the whole investigation of quantity; arithmetic, absolute quantity, and music, relative quantity.
And once more, inasmuch as part of “size” is in a state of rest and stability, and another part in motion and revolution, two other sciences in the same way will accurately treat of “size,” geometry the part that abides and is at rest, astronomy that which moves and revolves.
…In Plato’s Republic, when the interlocutor of Socrates appears to bring certain plausible reasons to bear upon the mathematical sciences, to show that they are useful to human life; arithmetic for reckoning, distributions, contributions, exchanges, and partnerships, geometry for sieges, the founding of cities and sanctuaries, and the partition of land, music for festivals, entertainment, and the worship of the gods, and the doctrine of the spheres, or astronomy, for farming, navigation and other undertakings, revealing beforehand the proper procedure and suitable season, Socrates, reproaching him, says: “You amuse me, because you seem to fear that these are useless studies that I recommend; but that is very difficult, nay, impossible. For the eye of the soul, blinded and buried by other pursuits, is rekindled and aroused again by these and these alone, and it is better that this be saved than thousands of bodily eyes, for by it alone is the truth of the universe beheld.”
–Nichomachus of Gerasa, Introduction to Arithmetic