Socrates:And the second, or bottom number, is the result of an odd number times itself?
Boy:Yes.
Socrates:And an even number is two times one whole number? Boy:Of course.
Socrates:So if we use this even number twice in multiplication, as we have on top, we have two twos times two whole numbers?
Boy:Yes, Socrates.
Socrates:(nudges Meno) and therefore the top number is four times some whole number times that whole number again?
Boy: Yes, Socrates. Socrates: And this number on top has to be twice the number on the bottom, if the even over odd number we began with is to give us two when multiplied by itself, or squared, as we call it?
Boy:Yes, Socrates.
Socrates: And if the top number is four times some whole number, then a number half as large would have to be two times that same whole number?
Boy:Of course, Socrates.
Socrates:So the number on the bottom is two times that whole number, whatever it is?
Boy:Yes, Socrates.
Socrates:(standing)And if it is two times a whole number, then it must be an even number, must it not?
Boy:Yes.
Socrates: Then is cannot be a member of the group which has an odd number on the bottom, can it?
Boy:No, Socrates.
Socrates: So can it be a member of the ratios created by an even number divided by an odd number and then used as a root to create a square?
Boy: No, Socrates. And that must mean it can't be a member of the last group, doesn't it?
Socrates: Yes, my boy, although I don't see how we can continue calling you boy, since you have now won your freedom, and are far richer than I will ever be.
Boy: Are you sure we have proved this properly? Let me go over it again, so I can see it in my head.
Socrates:Yes, my boy, er, ah, sir.
Boy: We want to see if this square root of two we discovered the other day is a member of the rational numbers?
Socrates:Yes.
Boy: So we define the rational numbers as numbers made from the division into ratios of whole numbers, whether those whole numbers are even or odd.
Socrates:Yes.
Boy: We get four groups, even over even, which we don't use, odd over even, odd over odd, and even over odd.
Socrates:Continue.
Boy: We know the first number in the squared ratio cannot be odd because it must be twice the value of the second number, and therefore is must be an even number, two times a whole number. Therefore it cannot be a member of either of the next groups, because they both have whole numbers over odd numbers.
Socrates:Wonderful!
Boy:So we are left with one group, the evens over odds. Socrates:Yes.
Boy: When we square an even over odd ratio, the first number becomes even times even, which is two times two times some other whole number, which means it is four times the whole number, and this number must be double the second number, which is odd, as it was made of odd times odd. But the top number cannot be double some bottom odd number because the top number is four times some whole number, and the bottom number is odd--but a number which is four times another whole number, cannot be odd when cut in half, so an even number times an even number can never be double what you would get from any odd number times another odd number. . .
therefore none of these rational numbers, when multiplied times themselves, could possibly yield a ratio in which the top number was twice the bottom number. Amazing. We have proved that the square root of two is not a rational number. Fantastic!
(he continues to wander up and down the stage, reciting various portions of the proof to himself, looking up, then down, then all around. He comes to Meno)Boy: Do you see? It's so simple, so clear. This is really wonderful! This is fantastic!
Socrates: (lays an arm on Meno's arm) Tell him how happy you are for his new found thoughts, Meno, for you can easily tell he is not thinking at all of his newly won freedom and wealth.
Meno: I quite agree with you, son, the clarity of your reasoning is truly astounding. I will leave you here with Socrates, as I go to prepare my household. I trust you will both be happy for the rest of the day without my assistance.
[The party, the presentation of 10 years salary to the newly freed young man, is another story, as is the original story of the drawing in the sant the square with an area of two.]
