Meno
Plato (428–347 BC)
Plato's Meno in the Greek text edited by John Burnet.
Section 70
Section 71
Section 72
Section 73
Section 74
Section 75
Section 76
Section 77
Section 78
Section 79
Section 80
Section 81
Section 82
Section 83
[83] SOCRATES: Observe him while he recalls the steps in regular order. (To the Boy:) Tell me, boy, do you assert that a double space comes from a double line? Remember that I am not speaking of an oblong, but of a figure equal every way, and twice the size of this--that is to say of eight feet; and I want to know whether you still say that a double square comes from double line?
BOY: Yes.
SOCRATES: But does not this line become doubled if we add another such line here?
BOY: Certainly.
SOCRATES: And four such lines will make a space containing eight feet?
BOY: Yes.
SOCRATES: Let us describe such a figure: Would you not say that this is the figure of eight feet?
BOY: Yes.
SOCRATES: And are there not these four divisions in the figure, each of which is equal to the figure of four feet?
BOY: True.
SOCRATES: And is not that four times four?
BOY: Certainly.
SOCRATES: And four times is not double?
BOY: No, indeed.
SOCRATES: But how much?
BOY: Four times as much.
SOCRATES: Therefore the double line, boy, has given a space, not twice, but four times as much.
BOY: True.
SOCRATES: Four times four are sixteen--are they not?
BOY: Yes.
SOCRATES: What line would give you a space of eight feet, as this gives one of sixteen feet;--do you see?
BOY: Yes.
SOCRATES: And the space of four feet is made from this half line?
BOY: Yes.
SOCRATES: Good; and is not a space of eight feet twice the size of this, and half the size of the other?
BOY: Certainly.
SOCRATES: Such a space, then, will be made out of a line greater than this one, and less than that one?
BOY: Yes; I think so.
SOCRATES: Very good; I like to hear you say what you think. And now tell me, is not this a line of two feet and that of four?
BOY: Yes.
SOCRATES: Then the line which forms the side of eight feet ought to be more than this line of two feet, and less than the other of four feet?
BOY: It ought.
SOCRATES: Try and see if you can tell me how much it will be.
BOY: Three feet.
SOCRATES: Then if we add a half to this line of two, that will be the line of three. Here are two and there is one; and on the other side, here are two also and there is one: and that makes the figure of which you speak?
BOY: Yes.
SOCRATES: But if there are three feet this way and three feet that way, the whole space will be three times three feet?
BOY: That is evident.
SOCRATES: And how much are three times three feet?
BOY: Nine.
SOCRATES: And how much is the double of four?
BOY: Eight.
SOCRATES: Then the figure of eight is not made out of a line of three?
BOY: No.