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 traced not only in divine and dæmonish 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