**Socrates:** Boy, you see this square, 2 ft by 2 ft.

**Boy:** Yes.

**Socrates:** Do you know its area?

**Boy:** It is a 4 ft. square, Socrates.

**Socrates:** Very good. Can you draw the side of a square that would be double the area of this one?

**Boy:** Yes.

(Boy draws out one side of the square so that it is 4 ft. long.)

**Socrates:** I see you have made your side double the side of the original square. Now draw the new square that can be constructed on the side you have drawn.

(Boy draws a square that is 4 ft. by 4 ft.)

**Socrates:** What is the area of this square?

**Boy:** 16 ft. square.

**Socrates:** That’s four times the area of the first square, not double. We need a square that is half of this one. Let’s go back to the original 2 foot square and try again. (Socrates erases the 16 ft. square and redraws the original one.)

(Boy thinks for a moment and then draws a length that is 3 feet)

**Socrates:** The side of your square is now 3ft?

**Boy:** Yes, Socrates.

**Socrates:** Now draw a square on this new side and tell me the area of the new square.

**Boy:** It is 9 ft. square, Socrates. Still too big.

**Socrates:** Well, let’s go back to the original square and try yet again. (Socrates erases the 9 ft. square and redraws the original one.) What will be side of a square that is double in area of this one?

**Boy:** (looking puzzled, hesitates, scratches his head, etc and says in bewilderment): I don’t know, Socrates.

**Socrates:** We can at least say that you are better off than before when you thought that you knew the answer, but now at least you know that you don’t.

**Boy:** Yes, Socrates. You are right. I am better off.

**Socrates:** Let’s go back to your 16 ft. square and think.

(Socrates draws the 16 ft. square and divides the square into fourths.)

**Socrates:** We want a square that is half the area of 16 ft. square and equal to the sum of two of these inner squares.

**Boy:** Yes.

**Socrates:** Well, what should we do?

**Boy:** (he thinks, puzzled, and says) I told you, Socrates, I don’t know.

**Socrates:** I think you do. Draw the diagonal of one of the inner squares, Boy, and see if you see anything.

**Boy:** Like this?

**Socrates:** Yes. What do you see?

(Boy looks, deep in thought, thinks and then with a rush of pleasure and delight says)

**Boy:** O Socrates, I know, I know. (And he draws the other three diagonals.)

**Socrates:** Explain, Boy!

**Boy:** Yes. (With some excitement) The diagonals must all be the same length because the squares are the same and each diagonal cuts its own square in half, so the resulting square must be half the size the of the 16 ft. square and therefore is that very 8 ft. square we have been searching for.

**Socrates:** Well done!

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