Page 186 - The_secret_teachings_of_all_ages_Neat

AN INTRODUCTION TO THE PYTHAGOREAN THEORY OF
NUMBERS

(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 latter 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, is 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 pare having priority. Multitude is also 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.

Owing to the fragmentary condition of existing Pythagorean records, it is difficult to
arrive at exact definitions of terms. Before it is possible, however, to unfold the subject
further some light must he cast upon the meanings of the words number, monad, and one.

The monad signifies (a) the all-including ONE. The Pythagoreans called the monad the
"noble number, Sire of Gods and men." The monad also signifies (b) the sum of any
combination of numbers considered as a whole. Thus, the universe is considered as a
monad, but the individual parts of the universe (such as the planets and elements) are
monads in relation to the parts of which they themselves are composed, though they, in
turn, are parts of the greater monad formed of their sum. The monad may also be likened
(c) to the seed of a tree which, when it has grown, has many branches (the numbers). In
other words, the numbers are to the monad what the branches of the tree are to the seed of