Meno, second part | Page 9

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 these
numbers 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.
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