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Yet a few more words may be added, on the old theme, which was so

oft discussed in the Socratic circle, of which we, like Glaucon

and Adeimantus, would fain, if possible, have a clearer notion.

Like them, we are dissatisfied when we are told that the idea of

good can only be revealed to a student of the mathematical

sciences, and we are inclined to think that neither we nor they

could have been led along that path to any satisfactory goal. For

we have learned that differences of quantity cannot pass into

differences of quality, and that the mathematical sciences can

never rise above themselves into the sphere of our higher

thoughts, although they may sometimes furnish symbols and

expressions of them, and may train the mind in habits of

abstraction and self-concentration. The illusion which was

natural to an ancient philosopher has ceased to be an illusion to

us.