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.