went
much further than this. They drew what seems to us the extraordinary
conclusion that the world is _made of_ {36} numbers. At this point,
then, we reach the heart of the Pythagorean philosophy. Just as Thales
had said that the ultimate reality, the first principle of which
things are composed, is water, so now the Pythagoreans teach that the
first principle of things is number. Number is the world-ground, the
stuff out of which the universe is made.

In the detailed application of this principle to the world of things
we have a conglomeration of extraordinary fancies and extravagances.
In the first place, all numbers arise out of the unit. This is the
prime number, every other number being simply so many units. The unit
then is the first in the order of things in the universe. Again,
numbers are divided into odd and even. The universe, said the
Pythagoreans, is composed of pairs of opposites and contradictories,
and the fundamental character of these opposites is that they are
composed of the odd and even. The odd and even, moreover, they
identified with the limited and the unlimited respectively. How this
identification was made seems somewhat doubtful. But it is clearly
connected with the theory of bipartition. An even number can be
divided by two and therefore it does not set a limit to bipartition.
Hence it is unlimited. An odd number cannot be divided by two, and
therefore it sets a limit to bipartition. The limited and the
unlimited become therefore the ultimate principles of the universe.
The Limit is identified with the unit, and this again with the central
fire of the universe. The Limit is first formed and proceeds to draw
more and more of the unlimited towards itself, and to limit it.
Becoming limited, it becomes a definite "something," a thing. So the
formation of the {37} world of things proceeds. The Pythagoreans drew
up a list of ten opposites of which the universe is composed. They are
(1) Limited and unlimited, (2) odd and even, (3) one and many, (4)
right and left, (5) masculine and feminine, (6) rest and motion, (7)
straight and crooked, (8) light and darkness, (9) good and evil, (10)
square and oblong.

With the further development of the number-theory Pythagoreanism
becomes entirely arbitrary and without principle. We hear, for
example, that 1 is the point, 2 is the line, 3 is the plane, 4 is the
solid, 5 physical qualities, 6 animation, 7 intelligence, health,
love, wisdom. There is no princ