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.