SOCRATES: Then now let us divide the arts of which we were speaking into two kinds,--the arts which, like music, are less exact in their results, and those which, like carpentering, are more exact. PROTARCHUS: Let us make that division. SOCRATES: Of the latter class, the most exact of all are those which we just now spoke of as primary. PROTARCHUS: I see that you mean arithmetic, and the kindred arts of weighing and measuring. SOCRATES: Certainly, Protarchus; but are not these also distinguishable into two kinds? PROTARCHUS: What are the two kinds? SOCRATES: In the first place, arithmetic is of two kinds, one of which is popular, and the other philosophical. PROTARCHUS: How would you distinguish them? SOCRATES: There is a wide difference between them, Protarchus; some arithmeticians reckon unequal units; as for example, two armies, two oxen, two very large things or two very small things. The party who are opposed to them insist that every unit in ten thousand must be the same as every other unit. PROTARCHUS: Undoubtedly there is, as you say, a great difference among the votaries of the science; and there may be reasonably supposed to be two sorts of arithmetic. SOCRATES: And when we compare the art of mensuration which is used in building with philosophical geometry, or the art of computation which is used in trading with exact calculation, shall we say of either of the pairs that it is one or two? PROTARCHUS: On the analogy of what has preceded, I should be of opinion that they were severally two. SOCRATES: Right; but do you understand why I have discussed the subject? PROTARCHUS: I think so, but I should like to be told by you. SOCRATES: The argument has all along been seeking a parallel to pleasure, and true to that original design, has gone on to ask whether one sort of knowledge is purer than another, as one pleasure is purer than another. PROTARCHUS: Clearly; that was the intention. SOCRATES: And has not the argument in what has preceded, already shown that the arts have different provinces, and vary in their degrees of certainty? PROTARCHUS: Very true. SOCRATES: And just now did not the argument first designate a particular art by a common term, thus making us believe in the unity of that art; and then again, as if speaking of two different things, proceed to enquire whether the art as pursed by philosophers, or as pursued by non-philosophers, has more of certainty and purity? PROTARCHUS: That is the very question which the argument is asking. SOCRATES: And how, Protarchus, shall we answer the enquiry? PROTARCHUS: O Socrates, we have reached a point at which the difference of clearness in different kinds of knowledge is enormous. SOCRATES: Then the answer will be the easier.