^{1}This proem was the effect of his experience in music, as we shall presently make out. The means from whence every mean is taken are three, arithmetical, enharmonical, geometrical. Of these the first exceeds and is exceeded in number, the second in proportion, the third neither in number nor proportion. Plato therefore, desirous to show the harmony of the four elements in the soul, and harmonically also to explain the reason of that mutual concord arising from discording and jarring principles, undertakes to make out two middle terms of the soul in every interval, according to harmonical proportion. Thus in a musical octave there happen to be two middle distances, whose proportion we shall explain. As for the octaves, they keep a double proportion between their two extremes. For example, let the double arithmetical proportion be 6 and 12, this being the interval between the ὑπάτη μέσων and [p. 119] the νήτη διεζευγμένων; 6 therefore and 12 being the two extremes, the former note contains the number 6, and the latter 12. To these are to be added the intermediate numbers, to which the extremes must hold the proportion, the one of one and a third, and the other of one and a half. These are the numbers 8 and 9. For as 8 contains one and a third of 6, so 9 contains one and a half of 6; thus you have one extreme. The other is 12, containing 9 and a third part of 9, and 8 and half 8. These then being the numbers between 6 and 12, and the interval of the octave consisting of a diatessaron and diapente, it is plain that the number 8 belongs to mese, and the number 9 to paramese; which being so, it follows that hypate is to mese as paramese to nete of the disjunct tetrachords; for it is a fourth from the first term to the second of this proportion, and the same interval from the third term to the fourth. The same proportion will be also found in the numbers. For as 6 is to 8, so is 8 to 12; and as 6 is to 9, so is 8 to 12. For 8 is one and a third part of 6, and 12 of 9; while 9 is one and a half part of 6, and 12 of 8. What has been said may suffice to show how great was Plato's zeal and learning in the liberal sciences.