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And yet tame gregarious animals have all, with the exception of about two species, been already divided; for dogs are not properly to be counted among gregarious creatures.

Younger Socrates
No, they are not. But how shall we divide the two species?

As you and Theaetetus ought by rights to divide them, since you are interested in geometry.

Younger Socrates
How do you mean?

By the diameter, of course, and again by the diameter of the square of the diameter.1

Younger Socrates
What do you mean by that?

1 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 διάμετρος, sometimes, as just below, διάμετρος δυνάμει δίπους, or, more briefly, 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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