raced not only in divine and daemonish things, but in
human works and words everywhere, and in all works of art and in music."
They even linked their arithmetical views to morality, through the
observation that numbers never lie; that they are hostile to falsehood;
and that, therefore, truth belongs to their family: their fanciful
speculations led them to infer that in the limitless or infinite,
falsehood and envy must reign. From similar reasoning, they concluded
that the number one contained not only the perfect, but also the
imperfect; hence it follows that the most good, most beautiful, and most
true are not at the beginning, but that they are in the process of time
evolved. They held that whatever we know must have had a beginning, a
middle, and an end, of which the beginning and end are the boundaries or
limits; but the middle is unlimited, and, as a consequence, may be
subdivided _ad infinitum_. They therefore resolved corporeal existence
into points, as is set forth in their maxim that "all is composed of
points or spacial units, which, taken together, constitute a number."
Such being their ideas of the limiting which constitutes the extreme,
they understood by the unlimited the intermediate space or interval. By
the aid of these intervals they obtained a conception of space; for,
since the units, or monads, as they were also called, are merely
geometrical points, no number of them could produce a line, but by the
union of monads and intervals conjointly a line can arise, and also a
surface, and also a solid. As to the interval thus existing between
monads, some considered it as being mere aerial breath, but the orthodox
regarded it as a vacuum; hence we perceive the meaning of their absurd
affirmation that all things are produced by a vacuum. As it is not to be
overlooked that the monads are merely mathematical points, and have no
dimensions or size, substances actually contain no matter, and are
nothing more than forms.

[Sidenote: Pythagorean cosmogony.]

[Sidenote: Modern Pythagorisms in chemistry.]

The Pythagoreans applied these principles to account for the origin of
the world, saying that, since its very existence is an illusion, it
could not have any origin in time, but only seemingly so to human
thought. As to time itself, they regarded it as "existing only by the
distinction of a series of different moments, which, however, are again
restored to unity by the limiting moments." The diversity of relat