Socrates: Very well, on then, to even over odd. If we multiply
these numbers times themselves, what do we get, boy?

Boy: We will get a ratio of even over odd, Socrates.

Socrates: And could an even number be double an odd number?

Boy: Yes, Socrates.

Socrates: So, indeed, this could be where we find a number
such that when multiplied times itself yields an area of two?

Boy: Yes, Socrates. It could very well be in this group.

Socrates: So, the first, or top number, is the result of an
even number times itself?

Boy: Yes.

Socrates: And the second, or bottom number, is the result of
an odd number times itself?

Boy: Yes.

Socrates: And an even number is two times one whole number?

Boy: Of course.

Socrates: So if we use this even number twice in multiplication,
as we have on top, we have two twos times two whole numbers?

Boy: Yes, Socrates.

Socrates: (nudges Meno) and therefore the top number is four
times some whole number times that whole number again?

Boy: Yes, Socrates.

Socrates: And this number on top has to be twice the number
on the bottom, if the even over odd number we began with is to
give us two when multiplied by itself, or squared, as we call it?

Boy: Yes, Socrates.

Socrates: And if the top number is four times some whole number, then
a number half as large would have to be two times that same whole
number?

Boy: Of course, Socrates.

Socrates: So the number on the bottom is two times that whole number,
whatever it is?

Boy: Yes, Socrates.

Socrates: (standing) And if it is two times a whole number,
then it must be an even number, must it not?

Boy: Yes.

Socrates: Then is cannot be a member of the group which has
an odd number on the bottom, can it?

Boy: No, Socrates.

Socrates: So can it be a member of the ratios created by an
even number divided by an odd number and then used as a root
to create a square?

Boy: No, Socrates. And that must mean it can't be a member
of the last group, doesn't it?

Socrates: Yes, my boy, although I don't see how we can
continue calling you boy, since you have now won your freedom,
and are far richer than I will ever be.

Boy: Are you sure we have proved this properly? Let me go
over it again, so I can see it in my head.

Socrates: Yes, my boy, er, ah, sir.

Boy: We want to see if this square root of two we discovered
the other day is a member of the rational numbers?

Socrates: