Unknown

“ Socrates: I can ask more of no man, Meno, and I am certain that you will do well, and I hope I will give you no call to halt me in my saying if I should say too much, in which you would feel I was actually teaching the boy the answer to this riddle.
Meno: No, Socrates, I don't think I will have to call you on anything you might say today, for the most wondrously learned men of the group of Pythagoras have spent many hours, weeks, and even months and years toiling in their manner to arrive at the mystic solutions to the puzzles formed by the simple squares with which we worked the other day. ”

“ It suits me well, Socrates.
Socrates: Then you know what odd and even are, boy?
Boy: Yes, shall I tell you?
Socrates: Please do. I would love to hear what they teach.
Boy: (the boy recites) A number can only be odd or even if it is a whole number, that is has no parts but only wholes of what it measures. Even numbers are special in that they have only whole twos in them, with no ones left over, while odd numbers always have a one left over when all the twos are taken out.
Socrates: An interesting, and somewhat effective definition.
Do you agree, Meno.
Meno: Yes, Socrates. Please continue. ”

“ Well, the easiest thing which comes to mind is to wager all those dinners you won from me the other day.
Socrates: Very well, so be it.
Meno: Now Socrates, since you are my friend, I must give you this friendly warning: you know that the Pythagoreans jealously guard their secrets with secret meetings, protected by secret handshakes, secret signs, passwords, and all that, do you not?
Socrates: I have heard as much, friend Meno.
Meno: Then be sure that they will seek revenge upon you for demystifying the ideas and concepts which they worked so long and hard and secretly to create and protect ”

“ 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. ”

“ 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. ”

“ Meno: You suppose correctly. I will feed you for a week of partying, and dress you in the finest garments, while you are introduced to the finest ladies and gentlemen of Athens, from whom you are free to select for your interests as friends, business partners, social acquaintances, connections, and perhaps even a wife, should you find someone you like for that. ”

“ 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. ”