31. For it was not out of vain-glory, to boast his skill in the mathematical sciences, that Plato inserted in a treatise of natural philosophy this discourse of harmonical and arithmetical medieties, but believing them both apt and convenient to demonstrate the structure and composition of the soul. For some there are who seek these proportions in the swift motions of the spheres of the planets; others rather in the distances, others in the magnitude of the stars; others, more accurate and nice in their enquiry, seek for the same proportions in the diameters of the epicycles; as if the Supreme Architect, for the sake of these, had adapted the soul, divided into seven parts, to the celestial bodies. Many also there are, who hither transfer the inventions of the Pythagoreans, tripling the distances of bodies from the middle. This is done by placing the unit next the fire; three next the Antichthon, or earth which is opposite to our earth; nine next the Earth; 27 next the Moon; 81 next to Mercury; 243 upon Venus; and 729 upon the Sun. The last (729) is both a tetragonal and cubical number, whence it is, that they also call the sun a tetragon and a cube. By this way of tripling they also reduce the other stars to proportion. But these people may be thought to dote and to wander very much from reason, if there be any use of geometrical demonstration, since by their mistakes we find that the most probable proofs proceed from thence; and although geometers do not always make out their positions exactly, yet they approach the nearest to truth when they say that the diameter of the sun, compared with the diameter of the earth, bears the proportion of 12 to 1; while the diameter of the earth to that of the moon carries a triple proportion. And for that which appears to be the least of the fixed stars, the diameter of it is no less than the third part of the diameter of the earth, and the whole globe of the earth to the whole globe of the moon is as twenty-seven to one. The diameters of Venus and the earth bear a duple, the globes or spheres of both an octave proportion. The width of the shadow which causes an eclipse holds a triple proportion to the diameter of the moon; and the deviation of the moon from the middle of the signs, either to the one or the other side, is a twelfth part. Her positions as to the sun, either in triangular or quadrangular distances, give her the form when she appears as in the first quarter and gibbous; but when she comes to be quite round, that is, when she has run through half the signs, she then makes (as it were) a kind of diapason harmony with six notes. But in regard the motions of the sun are slowest when he arrives at the solstices, and swiftest when he comes to the equinoxes, by which he takes from the day or adds to the night, the proportion holds thus. For the first thirty days after the winter solstice, he adds to the day a sixth part of the length whereby the longest night exceeds