Socrates Asks About Shapes

Meno, I am not teaching the boy anything. I am only asking him questions. He fancies that he knows how long a line is necessary in order to produce a shape of 8 square feet. But he really does not know. He only guesses that because the square is double, the line is double.

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?

SOCRATES: This line become doubled if we add another such line here. Four such lines will make a space containing 8 feet. Let us draw such a figure. There are these 4 divisions in the shape, each of which is equal to the figure of 4 feet. This is 4 times 4. Four times is not double but 4 times as much.

Therefore the double line, boy, has given a space, not twice, but four times as much. Four times four are 16.

A space of 8 feet, gives a line of 16 feet. The space of 4 feet is made from this half line.

A space of 8 feet is twice the size of this, and half the size of the other.

Such a space, then, will be made out of a line greater than this one, and less than that one.

This is a line of 2 feet and that of 4.

Then the line which forms the side of 8 feet should be more than this line of 2 feet, and less than the other of 4 feet. How much it will be?

If we add a half to this line of two, that will be the line of three.

Here are 2 and there is one. On the other side, here are 2 also and there is one. That makes the figure of which you speak.

The double of 4 is 8. Then the shape of eight is not made out of a line of 3.

But from what line?—tell me exactly. If you would rather not reckon, try and show me the line.