Halving a square

In Plato's dialog Meno, Socrates leads a slave boy to a discovery that the area of the large square is twice the area of the smaller one. There is a discussion as to the nature of the knowledge and origin of such facts.

The diagram on the right is suggestive of the proof. To make it rigorous, it may be necessary to stipulate commutativity and associativity of addition. Depending on your interpretation of the diagram, the only thing that may be needed is counting the number of equal right triangles.

Following is an excerpt from Plato's dialog Meno.

SOCRATES: Tell me, boy, is not this our square of four feet? (ABCD.) You understand?
BOY: Yes.
SOCRATES: Now we can add another equal to it like this? (BCEF.)
BOY: Yes.
SOCRATES: And a third here, equal to each of the others? (CEGH.)
BOY: Yes.
SOCRATES: And then we can fill in this one in the comer? (DCHJ.)
BOY: Yes.
SOCRATES: Then here we have four equal squares?
BOY: Yes.
SOCRATES: And how many times the size of the first square is the whole?
BOY: Four times.
SOCRATES: And we want one double the size. You remember?
BOY: Yes.
SOCRATES: Now does this line going from comer to comer cut each of these squares in half?
BOY: Yes.
SOCRATES: And these are four equal lines enclosing this area? (BEHD.)
BOY: They are.
SOCRATES: Now think. How big is this area?
BOY: I don't understand.
SOCRATES: Here are four squares. Has not each line cut off the inner half of each of them?
BOY: Yes.
SOCRATES: And how many such halves are there in this figure? (BEHD.)
BOY: Four.
SOCRATES: And how many in this one? (ABCD.)
BOY: Two.
SOCRATES: And what is the relation of four to two?
BOY: Double.
SOCRATES: How big is this figure then?
BOY: Eight feet.
SOCRATES: On what base?
BOY: This one.
SOCRATES: The line which goes from comer to comer of the square of four feet?
BOY: Yes.
SOCRATES: The technical name for it is 'diagonal'; so if we use that name, it is your personal opinion that the square on the diagonal of the original square is double its area.
BOY: That is so, Socrates.
SOCRATES: What do you think, Meno? Has he answered with any opinions that were not his own?
MENO: No, they were all his.
SOCRATES: Yet he did not know, as we agreed a few minutes ago.
MENO: True.
SOCRATES: But these opinions were somewhere in him, were they not?
MENO: Yes.
SOCRATES: So a man who does not know has in himself true opinions on a subject without having knowledge.
MENO: It would appear so.
SOCRATES: At present these opinions, being newly aroused, have a dream-like quality. But if the same questions are put to him on many occasions and in different ways, you can see that in the end he will have a knowledge on the subject as accurate as anybody's.
MENO: Probably.
SOCRATES: This knowledge will not come from teaching but from questioning. He will recover it for himself.
MENO: Yes.
SOCRATES: And the spontaneous recovery of knowledge that is in him is recollection, isn't it?
MENO: Yes.
SOCRATES: Either then he has at some time acquired the knowledge which he now has, or he has always possessed it. If he always possessed it, he must always have known; if on the other hand he acquired it at some previous time, it cannot have been in this life, unless somebody has taught him geometry. He will behave in the same way with all geometrical knowledge, and every other subject. Has anyone taught him all these? You ought to know, especially as he has been brought up in your household.
MENO: Yes, I know that no one ever taught him.
SOCRATES: And has he these opinions, or hasn't he?
MENO: It seems we can't deny it.
SOCRATES: Then if he did not acquire them in this life, isn't it immediately clear that he possessed and had learned them during some other period?
MENO: It seems so.
SOCRATES: When he was not in human shape?
MENO: Yes.
SOCRATES: If then there are going to exist in him, both while he is and while he is not a man, true opinions which can be aroused by questioning and turned into knowledge, may we say that his soul has been for ever in a state of knowledge? Clearly he always either is or is not a man.
MENO: Clearly.
SOCRATES: And if the truth about reality is always in our soul, the soul must be immortal, and one must take courage and try to discover-that is, to recollect what one doesn't happen to know, or (more correctly) remember, at the moment.
MENO: Somehow or other I believe you are right.
SOCRATES: I think I am. I shouldn't like to take my oath on the whole story, but one thing I am ready to fight for as long as I can, in word and act: that is, that we shall be better, braver and more active men if we believe it right to look for what we don't know than if we believe there is no point in looking because what we don't know we can never discover.
MENO: There too I am sure you are right.

(From a little different perspective the same episode is mentioned elsewhere. The problem also reappears in an engaging guide. Socrates argument shows that the Pythagorean theorem holds in a particular case of a right isosceles triangle. Rather surprisingly, the general case follows from such a restricted one.)

References

1. P.J.Davis and R.Hersh, The Mathematical Experience, Houghton Mifflin Company, Boston, 1981