r even ratios. What are they?

Boy: We know that both parts of the ratio have two in them.

Socrates: Excellent. See, Meno, how well he has learned his
lessons in school. His teacher must be proud, for I have taught
him nothing of this, have I?

Meno: No, I have not seen you teach it to him,
therefore he must have been exposed to it elsewhere.

Socrates: (back to the boy) And what have you learned about
ratios of even numbers, boy?

Boy: That both parts can be divided by two, to get the twos
out, over and over, until one part becomes odd.

Socrates: Very good. Do all school children know that, Meno?

Meno: All the ones who stay awake in class. (he stretches)

Socrates: So, boy, we can change the parts of the ratios,
without changing the real meaning of the ratio itself?

Boy: Yes, Socrates. I will demonstrate, as we do in class.
Suppose I use 16 and 8, as we did the other day. If I make a
ratio of 16 divided by 8, I can divide both the 16 and the 8 by
two and get 8 divided by 4. We can see that 8 divided by 4 is
the same as 16 divided by 8, each one is twice the other, as it
should be. We can then divide by two again and get 4 over 2, and
again to get 2 over 1. We can't do it again, so we say that this
fraction has been reduced as far as it will go, and everything
that is true of the other ways of expressing it is true of this.

Socrates: Your demonstration is effective. Can you divide
by other numbers than two?

Boy: Yes, Socrates. We can divide by any number which goes
as wholes into the parts which make up the ratio. We could have
started by dividing by 8 before, but I divided by three times,
each time by two, to show you the process, though now I feel
ashamed because I realize you are both masters of this,
and that I spoke to you in too simple a manner.

Socrates: Better to speak too simply, than in a manner in
which part or all of your audience gets lost, like the Sophists.

Boy: I agree, but please stop me if I get too simple.

Socrates: I am sure we can survive a simple explanation.
(nudges Meno, who has been gazing elsewhere) But back to your
simple proof: we know that a ratio of two even numbers can be
divided until reduced until one or both its parts are odd?

Boy: Yes, Socrates. Then it is a proper ratio.

Socrates: So we can eliminate one of our four groups, the
one where even was divided by even, and now we have odd/odd,
odd/even and even/odd?

Boy: Ye