Still, mathematics admit of other applications, as the Pythagoreans say, and we agree. There is a sister science of harmonical motion, adapted to the ear as astronomy is to the eye, and there may be other applications also. Let us inquire of the Pythagoreans about them, not forgetting that we have an aim higher than theirs, which is the relation of these sciences to the idea of good. The error which pervades astronomy also pervades harmonics. The musicians put their ears in the place of their minds. ‘Yes,’ replied Glaucon, ‘I like to see them laying their ears alongside of their neighbours’ faces—some saying, “That’s a new note,” others declaring that the two notes are the same.’ Yes, I said; but you mean the empirics who are always twisting and torturing the strings of the lyre, and quarrelling about the tempers of the strings; I am referring rather to the Pythagorean harmonists, who are almost equally in error. For they investigate only the numbers of the consonances which are heard, and ascend no higher,—of the true numerical harmony which is unheard, and is only to be found in problems, they have not even a conception. ‘That last,’ he said, ‘must be a marvellous thing.’ A thing, I replied, which is only useful if pursued with a view to the good.