How the principles of numbers were applied in the arts is not recorded, farther than what transpires in the works of Plato, whose doctrines were from the school of Pythagoras. In explaining the principle of beauty, as developed in the elements of the material world, he commences in the following words:—“But when the Artificer began to adorn the universe, he first of all figured with forms and numbers, fire and earth, water and air—which possessed, indeed, certain traces of the true elements, but were in every respect so constituted as it becomes anything to be from which Deity is absent. But we should always persevere in asserting that Divinity rendered them, as much as possible, the most beautiful and the best, when they were in a state of existence opposite to such a condition.” Plato goes on further to say, that these elementary bodies must have forms; and as it is necessary that every depth should comprehend the nature of a plane, and as of plane figures the triangle is the most elementary, he adopts two triangles as the originals or representatives of the isosceles and the scalene kinds. The first triangle of Plato is that which forms the half of the square, and is regulated by the number, 2; and the second, that which forms the half of the equilateral triangle, which is regulated by the number, 3; from various combinations of these, he formed the bodies of which he considered the elements to be composed. To these elementary figures I have already referred. Vitruvius, who studied architecture ages after the arts of Greece had been buried in the oblivion which succeeded her conquest, gives the measurements of various details of monuments of Greek art then existing. But he seems to have had but a vague traditionary knowledge of the principle of harmony and proportion from which these measurements resulted. He says—“The several parts which constitute a temple ought to be subject to the laws of symmetry; the principles of which should be familiar to all who profess the science of architecture. Symmetry results from proportion, which, in the Greek language, is termed analogy. Proportion is the commensuration of the various constituent parts with the whole; in the existence of which symmetry is found to consist. For no building can possess the attributes of composition in which symmetry and proportion are disregarded; nor unless there exist that perfect conformation of parts which may be observed in a well-formed human being.” After going at some length into details, he adds—“Since, therefore, the human figure appears to have been formed with such propriety, that the several members are commensurate with the whole, the artists of antiquity (meaning those of Greece at the period of her highest refinement) must be allowed to have followed the dictates of a judgment the most rational, when, transferring to works of art principles derived from nature, every part was so regulated as to bear a just proportion to the whole.