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[820a] and solid with solid?

Clinias
Absolutely.

Athenian
But supposing that some of them are neither absolutely nor moderately commensurable, some being commensurable and some not, whereas you regard them all as commensurable,—what do you think of your mental state with respect to them?

Clinias
Evidently it is a sorry state.

Athenian
Again, as regards the relation of line and surface to solid, or of surface and line to each other—do not all we Greeks imagine that these are somehow commensurable with one another? [820b]

Clinias
Most certainly.

Athenian
But if they cannot be thus measured by any way or means, while, as I said, all we Greeks imagine that they can, are we not right in being ashamed for them all, and saying to them, “O most noble Greeks, this is one of those ‘necessary’ things which we said1 it is disgraceful not to know, although there is nothing very grand in knowing such things.”

Clinias
Of course.

Athenian
In addition to these there are other matters, closely related to them, [820c] in which we find many errors arising that are nearly akin to the errors mentioned.

Clinias
What are they?

Athenian
Problems concerning the essential nature of the commensurable and the incommensurable. For students who are not to be absolutely worthless it is necessary to examine these and to distinguish the two kinds, and, by proposing such problems one to another, to compete in a game that is worthy of them,—for this is a much more refined pastime than draughts for old men. [820d]

Clinias
No doubt. And, after all, draughts and these studies do not seem to be so very far apart.

Athenian
I assert, then, Clinias, that these subjects must be learnt by the young; for they are, in truth, neither harmful nor hard, and when learnt by way of play they will do no damage at all to our State, but will do it good. Should anyone disagree, however, we must listen to him.

Clinias
Of course.

Athenian
Well then, if this is clearly the case, obviously we shall adopt these subjects; but if it seems clearly to be otherwise, we shall rule them out. [820e]

Clinias
Yes, obviously.

Athenian
Shall we not, then, lay these down as necessary subjects of instruction, so that there may be no gap in our code of laws? Yet we ought to lay them down provisionally—like pledges capable of redemption—apart from the rest of our constitution, in case they fail to satisfy either us who enact them or you for whom they are enacted.

Clinias
Yes, that is the right way to lay them down.

Athenian
Consider next whether or not we approve of the children learning astronomy.

Clinias
Just tell us your opinion.

Athenian
About this there is a very strange fact—indeed, quite intolerable.

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