Quadrivium

The quadrivium is the fourfold way of the mathematical sciences: arithmetic, the study of number in itself; geometry, of magnitude at rest; music, of number in ratio; astronomy, of magnitude in motion. The division is Pythagorean — Archytas already names the four as sister sciences — and Boethius, who coined the Latin word, transmitted it to the medieval schools as the fourfold path to wisdom.

Its charter is Republic VII, where Socrates prescribes exactly these studies for the philosopher-rulers: not for commerce or war, but because mathematical objects are the first realities the soul meets that do not change, and the soul that studies them is drawn 'from becoming to being.' The quadrivium is thus a ladder of purification and ascent: each science detaches the eye of the soul a little further from the senses and habituates it to the intelligible, until it is strong enough for dialectic and the vision of the Good itself.

Pythagoras is the fountainhead of the whole conception, and its richest ancient monuments are Nicomachus' Introduction to Arithmetic and the theological arithmetic of Iamblichus — the tradition gathered in English by Thomas Taylor's Theoretic Arithmetic, which opens with a defense of precisely this contemplative, non-mercantile mathematics.