Soc.

Now watch his progress in recollecting, by the proper use of memory. Tell me, boy, do you say we get the double space from the double line? The space I speak of is not long one way and short the other, but must be equal each way like this one, while being double its size—eight square feet. Now see if you still think we get this from a double length of line.

Boy.

I do.

Soc.

Well, this line is doubled, if we add here another of the same length?

Boy.

Certainly.

Soc.

And you say we shall get our eight-foot space from four lines of this length?

Boy.

Yes.

Soc.

Then let us describe the square, drawing four equal lines of that length. This will be what you say is the eight-foot figure, will it not?

Boy.

Certainly.

Soc.

And here, contained in it, have we not four squares, each of which is equal to this space of four feet?

Boy.

Yes.

Soc.

Then how large is the whole? Four times that space, is it not?

Boy.

It must be.

Soc.

And is four times equal to double?

Boy.

No, to be sure.

Soc.

But how much is it?

Boy.

Fourfold.

Soc.

Thus, from the double-sized line, boy, we get a space, not of double, but of fourfold size.

Boy.

That is true.

Soc.

And if it is four times four it is sixteen, is it not?

Boy.

Yes.

Soc.

What line will give us a space of eight feet? This one gives us a fourfold space, does it not?

Boy.

It does.

Soc.

And a space of four feet is made from this line of half the length?

Boy.

Yes.

Soc.

Very well; and is not a space of eight feet double the size of this one, and half the size of this other?

Boy.

Yes.

Soc.

Will it not be made from a line longer than the one of these, and shorter than the other?

Boy.

I think so.

Soc.

Excellent: always answer just what you think. Now tell me, did we not draw this line two feet, and that four?

Boy.

Yes.

Soc.

Then the line on the side of the eight-foot figure should be more than this of two feet, and less than the other of four?

Boy.

It should.

Soc.

Try and tell me how much you would say it is.

Boy.

Three feet.

Soc.

Then if it is to be three feet, we shall add on a half to this one, and so make it three feet? For here we have two, and here one more, and so again on that side there are two, and another one; and that makes the figure of which you speak.

Boy.

Yes.

Soc.

Now if it be three this way and three that way, the whole space will be thrice three feet, will it not?

Boy.

So it seems.

Soc.

And thrice three feet are how many?

Boy.

Nine.

Soc.

And how many feet was that double one to be?

Boy.

Eight.

Soc.

So we fail to get our eight-foot figure from this three-foot line.

Boy.

Yes, indeed.