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 a 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 masterin 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 even ratios, which we decided could be reduced in various manners to the other groups by dividing until one number of the ratio was no longer even; then we eliminated the two other groups which had odd numbers divided by either odd or even numbers, because the first or top number had to be twice the second or bottom number, and therefore could not be odd; this left the last group we are now to greet, even divided by odd.
Boy: Wonderfully put, Socrates. It is amazing how neatly you put an hour of thinking into a minute. Perhaps we can, indeed, put ten years of thinking into this one day. Please continue in this manner, if you know how it can be done.
Socrates: Would you have me continue, Meno? You know what shall have to happen if we solve this next group and do not find the square root of two in it.
Meno: Socrates, you are my friend, and my teacher, and a good companion. I will not shirk my duty to you or to this fine boy, who appears to be growing beyond my head, even as we speak. However, I still do not see that his head has reached the clouds wherein lie the minds of the Pythagoreans.
Socrates:Very well, on then, to even over odd.If we multiply thesenumbers 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.
