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Str. By the diameter, of course, and again by the diameter of the square of the diameter. The word diameter here denotes the diagonal of a square. The early Greek mathematicians worked out their arithmetical problems largely by geometrical methods (cf. Plat. Theaet. 147 D ff). The diagonal of the unit square (√2) was naturally of especial interest. It was called sometimes, as here simply ἡ διάμετρος, sometimes, as just below,ἡ διάμετρος ἡ δυνάμει δίπους, or, more briefly,ἡ διάμετρος δίπους. Given a square the side of which is the unit (i.e. one square foot), the length of the diagonal will be √2 and the square constructed with that diagonal as its side will contain two square feet. The length of the diagonal of this square will be √4=2 feet, and its area will be four square feet.
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Y. Soc. What do you mean by that?
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Str. Is the nature which our human race possesses related to walking in any other way than as the diameter which is the square root of two feet? There is here a play upon words. Man, being a two-footed (δίπους) animal, is compared to the diagonal of the unit square (√2,διάμετρος δίπους).
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Y. Soc. No.
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Str. And the nature of the remaining species, again, considered from the point of view of the square root, is the diameter of the square of our root, if it is the nature of twice two feet. i.e. the remaining species is four-footed. Our diameter is √2, and four is the area of the square constructed on the diagonal of the square which has √2 as its side. All this satirizes the tendency of contemporary thinkers to play with numbers.
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Y. Soc. Of course; and now I think I almost understand what you wish to make plain.
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Str. Socrates, do we see that besides this something else has turned up in these divisions of ours which would be a famous joke?
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Y. Soc. No. What is it?
