Socrates: So, boy, we can change the parts of the ratios,
without changing the real meaning of the ratio itself?
Boy: Yes, Socrates. I will demonstrate, as we do in class.
Suppose I use 16 and 8, as we did the other day. If I make a
ratio of 16 divided by 8, I can divide both the 16 and the 8 by
two and get 8 divided by 4. We can see that 8 divided by 4 is
the same as 16 divided by 8, each one is twice the other, as it
should be. 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.
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 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 whole numbers, whether those whole
numbers are even or odd.
Socrates: Yes.
Boy: We get four groups, even over even, which we don't use,
odd over even, odd over odd, and even over odd.
Socrates: Continue.
Boy: We know the first number in the squared ratio cannot be odd
because it must be twice the value of the second number,
and therefore is must be an even number, two times a whole number.
Therefore it cannot be a member of either of the next groups,
because they both have whole numbers over odd numbers.
Socrates: Wonderful!
Boy: So we are left with one group, the evens over odds.
Socrates: Yes.
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.]