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.

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.

Socrates: Let's try odd over even next, shall we?

Boy: Fine.

Socrates: What happens when you multiply an even number by an even number, what kind of number do you get, even or odd?

Boy: Even, of course. An even multiple of any whole number gives another even number.

Socrates: Wonderful, you have answered two questions, but we need only one at the moment. We shall save the other. So, with odd over even, if we multiply any of these times themselves, we well get odd times odd over even times even, and therefore odd over even, since odd times odd is odd and even of even is even.

Boy: Yes. A ratio of odd over even, when multiplied times itself, yields odd over even.

Socrates: And can our square root of two be in that group?

Boy: I don't know, Socrates. Have I failed?

Socrates: Oh, you know, you just don't know that you know.

Try this: after we multiply our number times itself, which the learned call "squaring" the number which is the root, we need to get a ratio in which the first or top number is twice as large as the second or bottom number. Is this much correct?

Boy: A ratio which when "squared" as you called it, yields an area of two, must then yield one part which is two times the other part. That is the definition of a ratio of two to one.

Socrates: So you agree that this is correct?

Boy: Certainly.

Socrates: Now if a number is to be twice as great as another, it must be two times that number?

Boy: Certainly.

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

Boy: Yes, Socrates.

Socrates: So, in our ratio we want to square to get two, the top number cannot be odd, can it?

Boy: No, Socrates. Therefore, the group of odd over even rational numbers cannot have the square root of two in it! Nor can the group ratios of odd numbers over odd numbers.

Socrates: Wonderful. We have just eliminated three of the four groups of rational numbers, first we eliminated the group of even over even numbers, then the ones with odd numbers divided by other numbers. However, these were the easier part, and we are now most of the way up the mountain, so we must rest and prepare to try even harder to conquer the rest, where the altitude is highest, and the terrain is rockiest. So let us sit and rest a minute, and look over what we have done, if you will.

Boy: Certainly, Socrates, though I am much invigorated by
the solution of two parts of the puzzle with one thought.
It was truly wonderful to see such simple effectiveness.
Are all great thoughts as simple as these, once you see them clearly?

Socrates: What do you say, Meno? Do thoughts get simpler as they get greater?

Meno: Well, it would appear that they do, for as the master of a great house, I can just order something be done, and it is; but if I were a master in a lesser house, I would have to watch over it much more closely to insure it got done. The bigger the decisions I have to make, the more help and advice I get in the making of them, so I would have to agree.

Socrates: Glad to see that you are still agreeable, Meno, though I think there are some slight differences in the way each of us view the simplicity of great thought. Shall we go on?

Meno: Yes, quite.

Boy: Yes, Socrates. I am ready for the last group, the ratios of even numbers divided by the odd, though, I cannot yet see how we will figure these out, yet, somehow I have confidence that the walls of these numbers shall tumble before us, as did the three groups before them.

Socrates: Let us review the three earlier groups, to prepare us for the fourth, and to make sure that we have not already broken the rules and therefore forfeited our wager. The four groups were even over