Soc. And how about the arts of reckoning and measuring as they are used in building and in trade when compared with philosophical geometry and elaborate computations—shall we speak of each of these as one or as two?

Pro. On the analogy of the previous example, I should say that each of them was two.

Soc. Right. But do you understand why I introduced this subject?

Pro. Perhaps; but I wish you would give the answer to your question.

Soc. This discussion of ours is now, I think, no less than when we began it, seeking a counterpart of pleasure, and therefore it has introduced the present subject and is considering whether there is one kind of knowledge purer than another, as one pleasure is purer than another.

Pro. That is very clear; it was evidently introduced with that object.

Soc. Well, had not the discussion already found in what preceded that the various arts had various purposes and various degrees of exactness?

Pro. Certainly.

Soc. And after having given an art a single name in what has preceded, thereby making us think that it was a single art, does not the discussion now assume that the same art is two and ask whether the art of the philosophers or that of the non-philosophers possesses the higher degree of clearness and purity?

Pro. Yes, I think that is just the question it asks.

Soc. Then what reply shall we make, Protarchus?

Pro. Socrates, we have found a marvelously great difference in the clearness of different kinds of knowledge.

Soc. That will make the reply easier, will it not?

Pro. Yes, to be sure; and let our reply be this, that the arithmetical and metrical arts far surpass the others and that of these the arts which are stirred by the impulse of the true philosophers are immeasurably superior in accuracy and truth about measures and numbers.

Soc. We accept that as our judgement, and relying upon you we make this confident reply to those who are clever in straining arguments—

Pro. What reply?

Soc. That there are two arts of arithmetic and two of measuring, and many other arts which, like these, are twofold in this way, but possess a single name in common.