ian would
seek to obtain from them true conceptions of equality or numerical
relations. How ridiculous then to look for these in the map of the
heavens, in which the imperfection of matter comes in everywhere as a
disturbing element, marring the symmetry of day and night, of months and
years, of the sun and stars in their courses. Only by problems can we
place astronomy on a truly scientific basis. Let the heavens alone, and
exert the intellect.

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.

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. 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