to be units.), taking
care that one shall continue one and not become lost in fractions.

That is very true.

Now, suppose a person were to say to them: O my friends, what are these
wonderful numbers about which you are reasoning, in which, as you say,
there is a unity such as you demand, and each unit is equal, invariable,
indivisible,--what would they answer?

They would answer, as I should conceive, that they were speaking of
those numbers which can only be realized in thought.

Then you see that this knowledge may be truly called necessary,
necessitating as it clearly does the use of the pure intelligence in the
attainment of pure truth?

Yes; that is a marked characteristic of it.

And have you further observed, that those who have a natural talent for
calculation are generally quick at every other kind of knowledge; and
even the dull, if they have had an arithmetical training, although they
may derive no other advantage from it, always become much quicker than
they would otherwise have been.

Very true, he said.

And indeed, you will not easily find a more difficult study, and not
many as difficult.

You will not.

And, for all these reasons, arithmetic is a kind of knowledge in which
the best natures should be trained, and which must not be given up.

I agree.

Let this then be made one of our subjects of education. And next, shall
we enquire whether the kindred science also concerns us?

You mean geometry?

Exactly so.

Clearly, he said, we are concerned with that part of geometry which
relates to war; for in pitching a camp, or taking up a position,
or closing or extending the lines of an army, or any other military
manoeuvre, whether in actual battle or on a march, it will make all the
difference whether a general is or is not a geometrician.

Yes, I said, but for that purpose a very little of either geometry or
calculation will be enough; the question relates rather to the greater
and more advanced part of geometry--whether that tends in any degree
to make more easy the vision of the idea of good; and thither, as I was
saying, all things tend which compel the soul to turn her gaze towards
that place, where is the full perfection of being, which she ought, by
all means, to behold.

True, he said.

Then if geometry compels us to view being, it concerns us; if becoming
only, it does not concern us?

Yes, that is what we assert.

Yet anybody who has the least acquaintance with geometr