Meno, second part | Page 8

as far as it will go, and everything that is
true of the other ways of expressing it is true of this.
Socrates: Your demonstration is effective. Can you divide by other
numbers than two?
Boy: Yes, Socrates. We can divide by any number which goes as
wholes into the parts which make up the ratio. We could have started
by dividing by 8 before, but I divided by three times, each time by two,
to show you the process, though now I feel ashamed because I realize
you are both masters of this, and that I spoke to you in too simple a
manner.
Socrates: Better to speak too simply, than in a manner in which part or
all of your audience gets lost, like the Sophists.
Boy: I agree, but please stop me if I get too simple.
Socrates: I am sure we can survive a simple explanation. (nudges Meno,
who has been gazing elsewhere) But back to your simple proof: we
know that a ratio of two even numbers can be divided until reduced
until one or both its parts are odd?
Boy: Yes, Socrates. Then it is a proper ratio.
Socrates: So we can eliminate one of our four groups, the one where
even was divided by even, and now we have odd/odd, odd/even and
even/odd?
Boy: Yes, Socrates.
Socrates: Let's try odd over even next, shall we?
Boy: Fine.
Socrates: What happens when you multiply an even number by an even
number, what kind of number do you get, even or odd?
Boy: Even, of course. An even multiple of any whole number gives
another even number.
Socrates: Wonderful, you have answered two questions, but we need
only one at the moment. We shall save the other. So, with odd over
even, if we multiply any of these times themselves, we well get odd

times odd over even times even, and therefore odd over even, since odd
times odd is odd and even of even is even.
Boy: Yes. A ratio of odd over even, when multiplied times itself, yields
odd over even.
Socrates: And can our square root of two be in that group?
Boy: I don't know, Socrates. Have I failed?
Socrates: Oh, you know, you just don't know that you know.
Try this: after we multiply our number times itself, which the learned
call "squaring" the number which is the root, we need to get a ratio in
which the first or top number is twice as large as the second or bottom
number. Is this much correct?
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.
Boy: Certainly, Socrates, though I am much invigorated by the solution
of two parts of the puzzle with one thought. It was truly wonderful to

see such simple effectiveness. Are all great thoughts as simple as these,
once you see them clearly?
Socrates: What do you say, Meno? Do thoughts get simpler as they get
greater?
Meno: Well, it would appear that they do, for as the master of a great
house, I can just order something be done, and it is; but if I were a
master in a lesser house, I would have to watch over it much more
closely to insure it got done. The bigger the decisions I have to make,
the more help and advice I get in the making of them, so I would have
to agree.
Socrates: Glad to see