(The following outline of Pythagorean mathematics is a paraphrase of the opening chapters of Thomas Taylor's Theoretic Arithmetic, the rarest and most important compilation of Pythagorean mathematical fragments extant).

The Pythagoreans declared arithmetic to be the mother of the mathematical sciences. This is proved by the fact that geometry, music and astronomy are dependent upon it but it is not dependent upon them. Thus, geometry may be removed but arithmetic will remain; but if arithmetic be removed, geometry is eliminated.

In the same manner music depends upon arithmetic, but the elimination of music affects arithmetic only by limiting one of its expressions. The Pythagoreans also demonstrated arithmetic to be prior to astronomy, for the later is dependent upon both geometry and music. The size, form, and motion of the celestial bodies is determined by the use of geometry their harmony and rhythm by the use of music. If astronomy be removed, neither geometry nor music is injured; but if geometry and music be eliminated, astronomy is destroyed. The priority of both geometry and music to astronomy is therefore established. Arithmetic, however, prior to all; it is primary and fundamental.

Pythagoras instructed his disciples that the science of mathematics is divided into two major parts. The first is concerned with the multitude, or the constituent parts of a thing, and the second with the magnitude, or the relative size or density of a thing.

Magnitude is divided into two parts - magnitude which is stationary and magnitude which is movable, the stationary part having priority. Multitude is divided into two parts, for it is related both to itself and to other things, the first relationship having priority. Pythagoras assigned the science of arithmetic to multitude related to itself, and the art of music to multitude related to other things. Geometry likewise was assigned to stationary magnitude, and spherics (used partly in the sense of astronomy) to movable magnitude. Both multitude and magnitude were circumscribed by the circumference of mind. The atomic theory has proved size to be the result of number, for a mass is made up of minute units though mistaken by the uninformed for a single simple substance.

(to be continued in next post...)