--corporeal or
incorporeal. All things are _numerable_, and can be counted. Moreover,
it is impossible to conceive a universe in which number is not to be
found. You could easily imagine a universe in which there is no
colour, or no sweet taste, or a universe in which nothing possesses
weight. But you cannot imagine a universe in which there is no number.
This is an inconceivable thought. Upon these grounds we should be
justified in concluding that number is an extremely important aspect
of things, and forms a fundamental pad of the framework of the world.
And it is upon this aspect of things that the Pythagoreans laid
emphasis.

They drew attention to proportion, order, and harmony as the dominant
notes of the universe. Now when we examine the ideas of proportion,
order, and harmony, we shall see that they are closely connected with
number. Proportion, for example, must necessarily {35} be expressible
by the relation of one number to another. Similarly order is
measurable by numbers. When we say that the ranks of a regiment
exhibit order, we mean that they are arranged in such a way that the
soldiers stand at certain regular distances from each other, and these
distances are measurable by numbers of feet or inches. Lastly,
consider the idea of harmony. If, in modern times, we were to say that
the universe is a harmonious whole, we should understand that we are
merely using a metaphor from music. But the Pythagoreans lived in an
age when men were not practised in thought, and they confused cosmical
harmony with musical harmony. They thought that the two things were
the same. Now musical harmony is founded upon numbers, and the
Pythagoreans were the first to discover this. The difference of notes
is due to the different numbers of vibrations of the sounding
instrument. The musical intervals are likewise based upon numerical
proportions. So that since, for the Pythagoreans, the universe is a
musical harmony, it follows that the essential character of the
universe is number. The study of mathematics confirmed the
Pythagoreans in this idea. Arithmetic is the science of numbers, and
all other mathematical sciences are ultimately reducible to numbers.
For instance, in geometry, angles are measured by the number of
degrees.

Now, as already pointed out, considering all these facts, we might
well be justified in concluding that number is a very important aspect
of the universe, and is fundamental in it. But the Pythagoreans