other day, when you thought that the square of area
nine was actually a square of area eight?

Boy: Oh yes, Socrates! And I am sorely ashamed, because I still
do not know enough to make sure I never make such an error again,
and therefore I know my virtue and rightness are lacking.

Socrates: They are not lacking so much that they cannot be improved,
are they boy?

Boy: I should hope and pray not.

Socrates: Well today, you are going to tell us some things
about that number, which when multiplied by itself gives us two.

Boy: I will tell you everything I know, or think I know,
Socrates, and hope that I am correct or can be corrected.

Socrates: To Meno, surely he is a fine boy, eh Meno?

Meno: Yes, I am proud to own him, but I don't see how he
can be smart enough to do the work today that would take a
Pythagorean monk ten years of cloistered life to accomplish.

Socrates: We shall see. Boy, you are doing fine. I think I
could even make a scholar of you, though I fear you might turn to
wine and women with your new found wealth, if you succeed, rather
than continue to polish the wit which should get you that reward.

Boy: I don't think I would want to spend that much time with
women or with wine, Socrates.

Socrates: You will find something, no doubt. So, back to
the number which when square gives us two. What can we say about
such a number? Is it odd or even? Well it would have to be a
whole number to be one of those, would it not, and we saw the
other day what happens to whole numbers when they are squared?
They give us 1,4,9 and 16 as square areas, did they not?

Boy: Yes, Socrates, though I remember thinking that there
should have been a number which would give eight, Socrates?

Socrates: I think we shall find one, if we keep searching.
Now, this number, do you remember if it had to be larger or
smaller than one?

Boy: Larger, Socrates. For one squared gives only an area
of one, and we need and area of two, which is larger.

Socrates: Good. And what of two?

Boy: Two gives a square of four, which is too large.

Socrates: Fine. So the square root of two is smaller than
the side two which is the root of four, and larger than the side
one which yields one?

Boy: Yes, Socrates.

Socrates: (Turning to Meno) So now he is as far as most of
us get in determining the magnitude of the square root of two?
And getting farther is largely a matter of guesswork, is it not?

Men