The quadrivium (four ways) was the medieval course in mathematics, consisting of **arithmetic, music, geometry, and astronomy**. The Latin term “quadrivium” comes from an educational course for monasteries created by Boethius (480-524), but the Pythagoreans had viewed mathematics as a four-fold discipline for at least a thousand years before Boethius

Medieval mathematics lacked algebra and analytic geometry. Thus, concepts could not be taught using equations or coordinate graphing.

Students of Pythagoras classified Arithmetic and Music as the study of discrete (or “indivisible”: a-tomos) mathematical entities, and Geometry and Astronomy as dealing with continuous (or “divisible”: tomos) mathematical entities. The music and astronomy were, of course, not what is taught under those headings in a modern university:

“Music” --or music theory, if you will-- was the study of ratios. Sounds of different pitch had long been associated with strings of different length. There was no theory of a sound “wave” and thus no proof that pitch equates with a specific frequency of vibrating air. It was known, however, that a vibrating string, divided in half, produced a tone one octave higher: and the relations of different lengths and tones were catalogued in “scales” and examined mathematically.

“Astronomy”, similarly, was purely the study of motion. The notion that the objects in the sky ---“cislunar” objects, objects above the moon-- were physical bodies, with mass and subject to physical forces, had to wait for Kepler, Galileo and Newton.

Thus, using the nomenclature of “dimension”:

** arithmetic** was one dimensional, **music** was two-dimensional (pitch was conceptualized as the ratio or relation of two quantities),
**geometry** was three-dimensional, and
**astronomy** added a fourth dimension of time (movement).