ble and the other to the
intelligible, and then compare the subdivisions in respect of their
clearness and want of clearness, and you will find that the first
section in the sphere of the visible consists of images. And by images I
mean, in the first place, shadows, and in the second place, reflections
in water and in solid, smooth and polished bodies and the like: Do you
understand?

Yes, I understand.

Imagine, now, the other section, of which this is only the resemblance,
to include the animals which we see, and everything that grows or is
made.

Very good.

Would you not admit that both the sections of this division have
different degrees of truth, and that the copy is to the original as the
sphere of opinion is to the sphere of knowledge?

Most undoubtedly.

Next proceed to consider the manner in which the sphere of the
intellectual is to be divided.

In what manner?

Thus:--There are two subdivisions, in the lower of which the soul uses
the figures given by the former division as images; the enquiry can only
be hypothetical, and instead of going upwards to a principle descends
to the other end; in the higher of the two, the soul passes out of
hypotheses, and goes up to a principle which is above hypotheses, making
no use of images as in the former case, but proceeding only in and
through the ideas themselves.

I do not quite understand your meaning, he said.

Then I will try again; you will understand me better when I have made
some preliminary remarks. You are aware that students of geometry,
arithmetic, and the kindred sciences assume the odd and the even and the
figures and three kinds of angles and the like in their several branches
of science; these are their hypotheses, which they and every body are
supposed to know, and therefore they do not deign to give any account of
them either to themselves or others; but they begin with them, and go
on until they arrive at last, and in a consistent manner, at their
conclusion?

Yes, he said, I know.

And do you not know also that although they make use of the visible
forms and reason about them, they are thinking not of these, but of the
ideals which they resemble; not of the figures which they draw, but
of the absolute square and the absolute diameter, and so on--the forms
which they draw or make, and which have shadows and reflections in water
of their own, are converted by them into images, but they are really
seeking to behold the things themselv