Meno, second part | Page 6

Socrates: Do you remember when you tripped up and fell on your face
the other day, when you thought that the square of area nine was
actually a square of area eight?
Boy: Oh yes, Socrates! And I am sorely ashamed, because I still do not
know enough to make sure I never make such an error again, and
therefore I know my virtue and rightness are lacking.
Socrates: They are not lacking so much that they cannot be improved,
are they boy?
Boy: I should hope and pray not.
Socrates: Well today, you are going to tell us some things about that
number, which when multiplied by itself gives us two.
Boy: I will tell you everything I know, or think I know, Socrates, and
hope that I am correct or can be corrected.
Socrates: To Meno, surely he is a fine boy, eh Meno?
Meno: Yes, I am proud to own him, but I don't see how he can be smart
enough to do the work today that would take a Pythagorean monk ten
years of cloistered life to accomplish.
Socrates: We shall see. Boy, you are doing fine. I think I could even
make a scholar of you, though I fear you might turn to wine and
women with your new found wealth, if you succeed, rather than
continue to polish the wit which should get you that reward.
Boy: I don't think I would want to spend that much time with women or
with wine, Socrates.
Socrates: You will find something, no doubt. So, back to the number
which when square gives us two. What can we say about such a
number? Is it odd or even? Well it would have to be a whole number to
be one of those, would it not, and we saw the other day what happens
to whole numbers when they are squared? They give us 1,4,9 and 16 as
square areas, did they not?
Boy: Yes, Socrates, though I remember thinking that there should have
been a number which would give eight, Socrates?
Socrates: I think we shall find one, if we keep searching. Now, this
number, do you remember if it had to be larger or smaller than one?
Boy: Larger, Socrates. For one squared gives only an area of one, and
we need and area of two, which is larger.
Socrates: Good. And what of two?

Boy: Two gives a square of four, which is too large.
Socrates: Fine. So the square root of two is smaller than the side two
which is the root of four, and larger than the side one which yields one?
Boy: Yes, Socrates.
Socrates: (Turning to Meno) So now he is as far as most of us get in
determining the magnitude of the square root of two? And getting
farther is largely a matter of guesswork, is it not?
Meno: Yes, Socrates, but I don't see how he will do it.
Socrates: Neither does he. But I do. Watch! (turning to the boy) Now I
am going to tell you something you don't know, so Meno will listen
very closely to make sure he agrees that I can tell you. You know
multiplication, boy?
Boy: I thought I had demonstrated that, Socrates?
Socrates: So you have, my boy, has he not Meno?
Meno: Yes, Socrates, I recall he did the other day.
Socrates: And you know the way to undo multiplication?
Boy: It is called division, but I do not know it as well as multiplication,
since we have not studied it as long.
Socrates: Well, I will not ask you to do much division, but rather I will
ask you only whether certain answers may be called odd or even, and
the like. Does that suit you?
Boy: 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.
Socrates: Now boy, what do you get when you divide these odd and
even numbers by other odd and even numbers.
Boy: Sometimes you get whole numbers, especially when you divide
an even number by an even number, but odd numbers sometimes give

whole numbers, both odd and even, and sometimes they give numbers
which are not whole numbers, but have