Str. 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.
Y. Soc. No, they are not. But how shall we divide the two species?
Str. As you and Theaetetus ought by rights to divide them, since you are interested in geometry.
Y. Soc. How do you mean?
Str. By the diameter, of course, and again by the diameter of the square of the diameter. Footnote 1The 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.
Y. Soc. What do you mean by that?
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? Footnote 2There 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).
Y. Soc. No.
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. Footnote 3i.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.
Y. Soc. Of course; and now I think I almost understand what you wish to make plain.
Str. Socrates, do we see that besides this something else has turned up in these divisions of ours which would be a famous joke?
Y. Soc. No. What is it?
Str. Our human race shares the same lot and runs in the same heat as the most excellent and at the same time most easy-going race of creatures. Footnote 4The animal referred to is the pig. See P. Shorey, Classical Philology,1917, July, p. 308.
Y. Soc. Yes, I see that; it is a very queer result.
Str. Indeed? But is it not reasonable that they arrive last, who are the slowest?
Y. Soc. Yes, that is true.
Str. And do we fail to notice this further point, that the king appears in a still more ridiculous light, running along with the herd and paired in the race with the man of all others who is most in training for a life of careless ease? Footnote 5i.e. the swineherd, the pig belonging to γένει εὐχερεστάτῳ.
Y. Soc. Certainly he does.
Str. For now, Socrates, we have shown more clearly the truth of that which we said yesterday in our search for the sophist. Footnote 6See Plat. Soph. 227B.
Y. Soc. What was it?
Str. That this method of argument pays no more heed to the noble than to the ignoble, and no less honor to the small than to the great, but always goes on its own way to the most perfect truth.
Y. Soc. So it seems.
Str. Then shall I now, without waiting for you to ask me, guide you of my own accord along that shorter way referred to a moment ago that leads to the definition of the king?
Y. Soc. By all means.
Str. I say, then, that we ought at that time to have divided walking animals immediately into biped and quadruped, then seeing that the human race falls into the same division with the feathered creatures and no others, we must again divide the biped class into featherless and feathered, and when that division is made and the art of herding human beings is made plain, we ought to take the statesmanlike and kingly man and place him as a sort of charioteer therein, handing over to him the reins of the state, because that is his own proper science.