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 sand the square with an area of two.]