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 aërial 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 relations we find in the world they supposed to be occasioned by the bond of harmony. "Since the principles of things are neither similar nor congenerous, it is impossible for them to be brought into order except by the intervention of harmony, whatever may have been the manner in which it took place. Like and homogeneous things, indeed, would not have required harmony; but, as to the dissimilar and unsymmetrical, such must necessarily be held together by harmony if they are to be contained in a world of order." In this manner they confused together the ideas of number and harmony, regarding the world not only as a combination of contraries, but as an orderly and harmonical combination thereof. To particular numbers they therefore imputed great significance, asserting that "there are seven chords or harmonies, seven pleiads, seven vowels, and that certain parts of the bodies of animals change in the course of seven years." They carried to an extreme the numerical doctrine, assigning certain numbers as the representatives of a bird, a horse, a man. This doctrine may be illustrated by facts familiar to chemists, who, in like manner, attach significant numbers to the names of things. Taking hydrogen as unity, 6 belongs to carbon, 8 to oxygen, 16 to sulphur. Carrying those principles out, there is no substance,