Str. Good, Socrates! If you preserve this attitude of indifference to mere names, you will turn out richer in wisdom when you are old. But now we will, as you suggest, not trouble ourselves about the name; but do you see a way in which a man may show that the art of herding is twofold,
 and may thereby cause that which is now sought among a double number of things to be sought among half as many? Y. Soc. I am quite willing to try. I think one kind is the care of men, the other that of beasts. Str. You made the division with perfect willingness and courage. However, let us do our best not to fall again into your error. Y. Soc. What error? Str. We must not take a single small part, and set it off against many large ones, nor disregard species in making our division.
 On the contrary, the part must be also a species. It is a very fine thing to separate the object of our search at once from everything else, if the separation can be made correctly, and so, just now, you thought you had the right division and you hurried our discussion along, because you saw that it was leading towards man. But, my friend, it is not safe to whittle off shavings; it is safer to proceed by cutting through the middle, and in that way one is more likely to find classes.
 This makes all the difference in the conduct of research. Y. Soc. What do you mean by that, Stranger? Str. I must try to speak still more clearly, Socrates, out of regard for your capacity. Just at present it is impossible to make the matter entirely plain, but I will try to lay it before you a little more fully for the sake of clearness. Y. Soc. What is it, then, that you say we did wrongly in making our division just now? Str. It was very much as if, in undertaking to divide the human race into two parts,
 one should make the division as most people in this country do; they separate the Hellenic race from all the rest as one, and to all the other races, which are countless in number and have no relation in blood or language to one another, they give the single name barbarian ; then, because of this single name, they think it is a single species. Or it was as if a man should think he was dividing number into two classes by cutting off a myriad from all the other numbers, with the notion that he was making one separate class,
 and then should give one name to all the rest, and because of that name should think that this also formed one class distinct from the other. A better division, more truly classified and more equal, would be made by dividing number into odd and even, and the human race into male and female; as for the Lydians and Phrygians and various others they could be opposed to the rest and split off from them when it was impossible to find and separate two parts,
 each of which formed a class. Y. Soc. Very true; but that’s just the trouble, Stranger: how can we get a clearer knowledge of class and part, and see that they are not the same thing, but different? Str. Socrates, you most excellent young man, it is no small task you impose upon me. We have already strayed away from our subject more than we ought, and you wish us to wander still farther afield. So for the present let us return to our subject, as is proper; then we will go on the trail of this other matter by and by, when we have time.
 Only take very good care not to imagine that you ever heard me declare flatly— Y. Soc. What? Str. That class and part are separate from one another. Y. Soc. But what did you say? Str. That when there is a class of anything, it must necessarily be a part of the thing of which it is said to be a class; but there is no necessity that a part be also a class. Please always give this, rather than the other, as my doctrine. Y. Soc. I will do so. Str. Then please go on to the next point. Y. Soc. What is it? Str. That from which our present digression started. For I think it started when you were asked how the art of herding should be divided and said with great readiness that there were two kinds of living beings, the human race and a second one, a single class, comprising all the beasts. Y. Soc. True. Str. And it was clear to me at the time that you removed a part and then thought that the remainder was one class
 because you were able to call them all by the same name of beasts. Y. Soc. That is true, too. Str. But indeed, my most courageous young friend, perhaps, if there is any other animal capable of thought, such as the crane appears to be, or any other like creature, and it perchance gives names, just as you do, it might in its pride of self oppose cranes to all other animals, and group the rest, men included, under one head, calling them by one name, which might very well be that of beasts. Now let us try to be on our guard against all that sort of thing. Y. Soc. How can we guard against it? Str. By not dividing the whole class of living beings, that so we may avoid such errors. Y. Soc. Well, there is no need of dividing the whole. Str. No, certainly not, for it was in that way that we fell into our former error. Y. Soc. What do you mean? Str. That part of intellectual science which involves giving commands was a part of our animal-tending class, with especial reference to animals in herds, was it not? Y. Soc. Yes. Str. Well, even at that stage of our discussion all animals had already been divided into tame and wild.
 For if their nature admits of domestication they are called tame; if it does not, they are called wild. Y. Soc. Excellent. Str. But the science we are hunting for was, and is, to be sought among tame creatures, more specifically creatures in herds. Y. Soc. Yes. Str. Let us, then, not make our division as we did before, with a view to all, nor in a hurry, with the idea that we may thus reach political science quickly,
 for that has already brought upon us the proverbial penalty. Y. Soc. What penalty? Str. The penalty of having made less speed, because we made too much haste and did not make our division right. Y. Soc. And it was a good thing for us, Stranger Str. I do not deny it. So let us begin again and try to divide the art of tending animals in common; for perhaps the information you desire so much will come to you in the ordinary course of our conversation better than by other means. Tell me— Y. Soc. What? Str. Whether, as I suppose, you have often heard people speak of this,—
 for I know you never actually saw the preserves of fish in the Nile and in the ponds of the Persian king. But perhaps you have noticed the like in fountain-pools. Y. Soc. Yes, I have often seen the fish in fountain-pools and have heard many tales of those foreign preserves. Str. And surely, even if you have not wandered over the plains of Thessaly , you have heard of goose-farms and crane-farms there and you believe that they exist. Y. Soc. Yes, of course. Str. The reason why I asked you all these questions is that the rearing of flocks is in part aquatic and in part an affair of the dry land. Y. Soc. Yes, that is true. Str. Then do you agree that we ought to divide the art of tending animals in common into corresponding parts, assigning one part of it to each of these two, and calling one the art of aquatic-herding and the other the art of land-herding? Y. Soc. Yes, I agree. Str. And surely we shall not have to ask
 to which of these two arts kingship belongs, for that is clear to everyone. Y. Soc. Of course. Str. Anybody could doubtless make a division of the art of tending herds on land. Y. Soc. What would the division be? Str. Into the tending of flying and walking animals. Y. Soc. Very true. Str. And statesmanship is to be sought in connection with walking animals, is it not? Any fool, so to speak, would believe that, don’t you think? Y. Soc. Of course. Str. And the art of tending animals that walk must, like an even number, be divided in half. Y. Soc. Evidently. Str. And now I think I see two paths leading in that direction in which our argument has started: the quicker way, by separating a relatively small part and a larger, and the other way, which is more in accord with what we said a while ago about the need of making the division as nearly in the middle as we can, but is longer. So we can proceed by whichever of the two we wish. Y. Soc. Can we not go by both? Str. Not by both at once, silly boy; but obviously we can take them in turn. Y. Soc. Then I choose both in turn. Str. That is easy enough, since we have but a short distance to go. At the beginning, certainly, or middle of our journey it would have been hard to comply with your demand. But now, since this is your wish, let us go first by the longer way, for we are fresher now and shall get along on it more easily. So attend to the division. Y. Soc. Go on. Str. The tame walking animals which live in herds are divided by nature into two classes. Y. Soc. How by nature? Str. Because one class is naturally without horns, and the other has horns. Y. Soc. That is obvious. Str. Now divide the art of tending herds of walking animals into two parts, assigning one to each class of animals; and define the parts, for if you try to give them names, the matter will become needlessly complicated. Y. Soc. How shall I speak of them then? Str. In this way: say that the science which tends herds of walking animals is divided into two parts, one of which is assigned to the horned portion of the herd, the other to the hornless portion. Y. Soc. Assume that I have said that;
 for you have made it perfectly clear. Str. And furthermore our king is very clearly the herdsman of a herd devoid of horns. Y. Soc. Of course; that is evident. Str. Let us then try to break up this herd and give the king the part that belongs to him. Y. Soc. Very well. Str. Shall we make our division on the basis of having or not having cloven hoofs, or on that of mixing or not mixing the breed? You know what I mean. Y. Soc. No. What is it? Str. Why, I mean that horses and asses can breed from each other. Y. Soc. Oh yes. Str. But the rest of the herd of hornless tame animals cannot cross the breed. Y. Soc. That is true, of course. Str. Well then, does the statesman appear to have charge of a kind that mixes or of one that does not mix the breed? Y. Soc. Evidently of one that is unmixed. Str. So I suppose we must proceed as we have done heretofore and divide this into two parts. Y. Soc. Yes, we must. 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. 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. 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? 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 ). 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. 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. 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. The 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? i.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. See 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.