Socrates: I mean to say, that if arithmetic, mensuration, and weighing be taken away from any art, that which remains will not be much.
Protarchus: Not much, certainly.
Socrates: The rest will be only conjecture, and the better use of the senses which is given by experience and practice, in addition to a certain power of guessing, which is commonly called art, and is perfected by attention and pains.
Protarchus: Nothing more, assuredly.
Socrates: Music, for instance, is full of this empiricism; for sounds are harmonized, not by measure, but by skilful conjecture; the music of the flute is always trying to guess the pitch of each vibrating note, and is therefore mixed up with much that is doubtful and has little which is certain.
Protarchus: Most true.
Socrates: And the same will be found to hold good of medicine and husbandry and piloting and generalship.
Protarchus: Very true.
Socrates: The art of the builder, on the other hand, which uses a number of measures and instruments, attains by their help to a greater degree of accuracy than the other arts.
Protarchus: How is that?
Socrates: In shipbuilding and house-building, and in other branches of the art of carpentering, the builder has his rule, lathe, compass, line, and a most ingenious machine for straightening wood.
Protarchus: Very true, Socrates.
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?