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 over 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.