By Robert Sherrick Brumbaugh

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Let it have the ratio which C has to D. AJathematical Imagination the side; (VIII. I I) therefore also, as the square on A is to the square on B, so is the square on C to the square on D. "Next, as the square on A is to the square on B, so let the square on C be to the square on D; I say that A is commensurable in length with B. "For since, as the square on A is to the square on B, so is the square on C to the square on D, while the ratio of the square on A to the square on B is duplicate of the ratio of A to B, and the ratio of the square on C to the square on D is duplicate of the ratio of C to D, therefore also, as A is to B, so is C to D.

I shall only ask him, and not teach him, and he shall share the enquiry with me: and do you watch and see if you find me telling or explaining anything to him, instead of eliciting his opinion. Tell me, boy, is not this a square of four feet which I have drawn? [ABED} Fig. 9] BOY: Yes. : And now I add another square equal to the former one? [BEFC} Fig. 9] BOY: Yes. : And a third, which is equal to either of them? [DEHG J Fig. 9] Plato's Mathematical Imagination 26 BOY: Yes. : Suppose that we fill up the vacant corner?

28 There is no reason why Socrates, if he had the more general problem in mind, should EXAMPLES FROM PURE MATHEMATICS OF METHODS AND CLASS-RELATIONS 35 not have presented to Meno tIle most intuitively evident special case from a cOlnplete demonstration which Meno could not have understood. This suggestion combines the merits of Benecke's insistence on consistency of character and economy of imagery, and the suggestion of the second group of interpreters, that some more general theorem is required to make the example mathematically interesting or meaningful.