ally weak. Could they have been united
in a powerful federation by the aid of some political or religious bond,
they might have exerted a singular influence on the rising fortunes of
Rome, and thereby on humanity.

[Sidenote: Pythagoras asserts that number is the first principle.]

The fundamental dogma of the Pythagoreans was that "number is the
essence or first principle of things." This led them at once to the
study of the mysteries of figures and of arithmetical relations, and
plunged them into the wildest fantasies when it took the absurd form
that numbers are actually things.

The approval of the doctrines of Pythagoras so generally expressed was
doubtless very much due to the fact that they supplied an intellectual
void. Those who had been in the foremost ranks of philosophy had come to
the conclusion that, as regard external things, and even ourselves, we
have no criterion of truth; but in the properties of numbers and their
relations, such a criterion does exist.

[Sidenote: Pythagorean philosophy.]

It would scarcely repay the reader to pursue this system in its details;
a very superficial representation of it is all that is necessary for our
purpose. It recognizes two species of numbers, the odd and even; and
since one, or unity, must be at once both odd and even, it must be the
very essence of number, and the ground of all other numbers; hence the
meaning of the Pythagorean expression, "All comes from one;" which also
took form in the mystical allusion, "God embraces all and actuates all,
and is but one." To the number ten extraordinary importance was imputed,
since it contains in itself, or arises from the addition of, 1, 2, 3,
4--that is, of even and odd numbers together; hence it received the name
of the grand tetractys, because it so contains the first four numbers.
Some, however, assert that that designation was imposed on the number
thirty-six. To the triad the Pythagoreans likewise attached much
significance, since it has a beginning, a middle, and an end. To unity,
or one, they gave the designation of the even-odd, asserting that it
contained the property both of the even and odd, as is plain from the
fact that if one be added to an even number it becomes odd, but if to an
odd number it becomes even. They arranged the primary elements of nature
in a table of ten contraries, of which the odd and even are one, and
light and darkness another. They said that "the nature and energy of
number may be t