Another mode of amusing them is to distribute vessels, sometimes of gold, brass, silver, and the like, intermixed with one another, sometimes of one metal only; as I was saying they adapt to their amusement the numbers in common use, and in this way make more intelligible to their pupils the arrangements and movements of armies and expeditions, and in the management of a household they make people more useful to themselves, and more wide awake; and again in measurements of things which have length, and breadth, and depth, they free us from that natural ignorance of all these things which is so ludicrous and disgraceful.
CLEINIAS: What kind of ignorance do you mean?
ATHENIAN: O my dear Cleinias, I, like yourself, have late in life heard with amazement of our ignorance in these matters; to me we appear to be more like pigs than men, and I am quite ashamed, not only of myself, but of all Hellenes.
CLEINIAS: About what?
Say, Stranger, what you mean.
ATHENIAN: I will; or rather I will show you my meaning by a question, and do you please to answer me: You know, I suppose, what length is?
CLEINIAS: Certainly.
ATHENIAN: And what breadth is?
CLEINIAS: To be sure.
ATHENIAN: And you know that these are two distinct things, and that there is a third thing called depth?
CLEINIAS: Of course.
ATHENIAN: And do not all these seem to you to be commensurable with themselves?
CLEINIAS: Yes.
ATHENIAN: That is to say, length is naturally commensurable with length, and breadth with breadth, and depth in like manner with depth?
CLEINIAS: Undoubtedly.
ATHENIAN: But if some things are commensurable and others wholly incommensurable, and you think that all things are commensurable, what is your position in regard to them?
CLEINIAS: Clearly, far from good.
ATHENIAN: Concerning length and breadth when compared with depth, or breadth and length when compared with one another, are not all the Hellenes agreed that these are commensurable with one another in some way?
CLEINIAS: Quite true.
ATHENIAN: But if they are absolutely incommensurable, and yet all of us regard them as commensurable, have we not reason to be ashamed of our compatriots; and might we not say to them: O ye best of Hellenes, is not this one of the things of which we were saying that not to know them is disgraceful, and of which to have a bare knowledge only is no great distinction?
CLEINIAS: Certainly.
ATHENIAN: And there are other things akin to these, in which there spring up other errors of the same family.
CLEINIAS: What are they?
ATHENIAN: The natures of commensurable and incommensurable quantities in their relation to one another.

A man who is good for anything ought to be able, when he thinks, to distinguish them; and different persons should compete with one another in asking questions, which will be a far better and more graceful way of passing their time than the old man's game of draughts.
CLEINIAS: I dare say; and these pastimes are not so very unlike a game of draughts.
ATHENIAN: And these, as I maintain, Cleinias, are the studies which our youth ought to learn, for they are innocent and not difficult; the learning of them will be an amusement, and they will benefit the state.
If any one is of another mind, let him say what he has to say.
CLEINIAS: Certainly.
ATHENIAN: Then if these studies are such as we maintain, we will include them; if not, they shall be excluded.
CLEINIAS: Assuredly: but may we not now, Stranger, prescribe these studies as necessary, and so fill up the lacunae of our laws?
ATHENIAN: They shall be regarded as pledges which may be hereafter redeemed and removed from our state, if they do not please either us who give them, or you who accept them.
CLEINIAS: A fair condition.
ATHENIAN: Next let us see whether we are or are not willing that the study of astronomy shall be proposed for our youth.
CLEINIAS: Proceed.
ATHENIAN: Here occurs a strange phenomenon, which certainly cannot in any point of view be tolerated.
CLEINIAS: To what are you referring?
ATHENIAN: Men say that we ought not to enquire into the supreme God and the nature of the universe, nor busy ourselves in searching out the causes of things, and that such enquiries are impious; whereas the very opposite is the truth.
CLEINIAS: What do you mean?
ATHENIAN: Perhaps what I am saying may seem paradoxical, and at variance with the usual language of age.

But when any one has any good and true notion which is for the advantage of the state and in every way acceptable to God, he cannot abstain from expressing it.
CLEINIAS: Your words are reasonable enough; but shall we find any good or true notion about the stars?
ATHENIAN: My good friends, at this hour all of us Hellenes tell lies, if I may use such an expression, about those great Gods, the Sun and the Moon.
CLEINIAS: Lies of what nature?
ATHENIAN: We say that they and divers other stars do not keep the same path, and we call them planets or wanderers.
CLEINIAS: Very true, Stranger; and in the course of my life I have often myself seen the morning star and the evening star and divers others not moving in their accustomed course, but wandering out of their path in all manner of ways, and I have seen the sun and moon doing what we all know that they do.
ATHENIAN: Just so, Megillus and Cleinias; and I maintain that our citizens and our youth ought to learn about the nature of the Gods in heaven, so far as to be able to offer sacrifices and pray to them in pious language, and not to blaspheme about them.
CLEINIAS: There you are right, if such a knowledge be only attainable; and if we are wrong in our mode of speaking now, and can be better instructed and learn to use better language, then I quite agree with you that such a degree of knowledge as will enable us to speak rightly should be acquired by us.

And now do you try to explain to us your whole meaning, and we, on our part, will endeavour to understand you.
ATHENIAN: There is some difficulty in understanding my meaning, but not a very great one, nor will any great length of time be required.

And of this I am myself a proof; for I did not know these things long ago, nor in the days of my youth, and yet I can explain them to you in a brief space of time; whereas if they had been difficult I could certainly never have explained them all, old as I am, to old men like yourselves.
CLEINIAS: True; but what is this study which you describe as wonderful and fitting for youth to learn, but of which we are ignorant?
Try and explain the nature of it to us as clearly as you can.
ATHENIAN: I will.