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. *531* 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. {cv} '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. All these sciences are the prelude of the strain, and are profitable if they are regarded in their natural relations to one another. 'I dare say, Socrates,' said Glaucon; 'but such a study will be an endless business.' What study do you mean--of the prelude, or what? For all these things are only the prelude, and you surely do not suppose that a mere mathematician is also a dialectician? 'Certainly not. *532* I have hardly ever known a mathematician who could reason.' And yet, Glaucon, is not true reasoning that hymn of dialectic which is the music of the intellectual world, and which was by us compared to the effort of sight, when from beholding the shadows on the wall we arrived at last at the images which gave the shadows? Even so the dialectical faculty withdrawing from sense arrives by the pure intellect at the contemplation of the idea of good, and never rests but at the very end of the intellectual world. And the royal road out of the cave into the light, and the blinking of the eyes at the sun and turning to contemplate the shadows of reality, not the shadows of an image only--this progress and gradual acquisition of a new faculty of sight by the help of the mathematical sciences, is the elevation of the soul to the contemplation of the highest ideal of being.