, Socrates.

Socrates: Do you agree with the way I told him this, Meno?
Does it violate our agreement?

Meno: You added -nal to the word ratio, just as we add -nal
to the French word "jour" to create the word journal which means
something that contains words of the "jour" or of today. So we
now have a word which means a number made from a ratio. This is
more than acceptable to me, Socrates. A sort of lesson in
linguistics, perhaps, but certainly not in mathematics. No, I do
not see that you have told him how to solve anything about the
square root of two, but thank you for asking. I give you your
journalistic license to do so.

Socrates: Good. Now boy, I need your attention. Please get
up and stretch, if it will help you stay and think for awhile.

Boy: (stretches only a little) I am fine, Socrates.

Socrates: Now think carefully, boy, what kind of ratios can
we make from even numbers and odd numbers?

Boy: We could make even numbers divided by odd numbers, and
odd numbers divided by even numbers.

Socrates: Yes, we could. Could we make any other kind?

Boy: Well... we could make even numbers divided by even
numbers, or odd numbers divided by odd.

Socrates: Very good. Any other kind?

Boy: I'm not sure, I can't think of any, but I might have to
think a while to be sure.

Socrates: (to Meno) Are you still satisfied.

Meno: Yes, Socrates. He knows even and odd numbers,
and ratios; as do all the school children his age.

Socrates: Very well, boy. You have named four kinds of
ratios: Even over odd, odd over even, even over even, odd over
odd, and all the ratios make numbers we call rational numbers.

Boy: That's what it looks like, Socrates.

Socrates: Meno, have you anything to contribute here?

Meno: No, Socrates, I am fine.

Socrates: Very well. Now, boy, we are off in search of more
about the square root of two. We have divided the rational numbers
into four groups, odd/even, even/odd, even/even, odd/odd?

Boy: Yes.

Socrates: And if we find another group we can include them.
Now, we want to find which one of these groups, if any, contains
the number you found the other day, the one which squared is two.

Would that be fun to try?

Boy: Yes, Socrates, and also educational.

Socrates: I think we can narrow these four groups down to
three, and thus make the search easier. Would you like that?

Boy: Certainly, Socrates.

Socrates: Let's take even ove