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: Yes, Socrates.