he
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 ship-building 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 us