From the Beginning to Plato

252 FROM THE BEGINNING TO PLATO
represents by straight lines. The essential content of the definition may be
paraphrased as follows:
V, def. 5. x:y :: z:w if and only if whenever a multiple m·x is greater than
(equal to or less than) a multiple n·y, m·z is greater than (equal to or less
than) n·w, and (V, def. 7) if for some m and n
and .
Euclid proceeds to prove a number of important laws of proportionality, e.g.,
alternation (V.16: if x:y :: z:w, then x:z :: y:w) using a strictly formal reduction
to the definition and to basic properties of multiplication and size comparison. In
Book VI Euclid applies these laws to plane geometric objects.
Geometry disappears at the end of Book VI when Euclid turns to arithmetic,
the subject of VII–IX. Logically these three books are completely independent of
the first six. In them Euclid uses a notion of proportionality specific to numbers,
i.e., positive integers, and proves for numbers laws of proportionality already
proven for magnitudes in Book V.
In Book X there is a kind of unification of arithmetic and geometry. Euclid
distinguishes between commensurable and incommensurable magnitudes, again
represented by straight lines, and proves:
X.5–8. Two magnitudes are commensurable if and only if they have the
ratio of a number to a number.
I shall briefly discuss the proof of these propositions at the end of section 5, but
in the immediate discussion I shall treat this equivalence as something which can
be taken for granted as well known to Greek mathematicians of the fourth and
perhaps even fifth century. The bulk of Book X is given over to an elaborate
classification of certain ‘irrational’ (alogos) straight lines. 4 Euclid proves the
‘irrationality’ of a number of straight lines, the most important being:
the medial, the side of a square equal to a rectangle with incommensurable
‘rational’ sides;
the binomial, the sum of two incommensurable ‘rational’ straight lines;
the apotome, the difference of two incommensurable ‘rational’ straight
lines.
Book XI covers a great deal of elementary solid geometry quite rapidly; by
contrast with his procedure in Books I—IV and VI, Euclid appears willing to use
proportionality whenever he thinks it simplifies his argumentation. Book XII is
characterized by the method used in establishing its principal results, the socalled
method of exhaustion. The method of exhaustion is, in fact, a rigorous
technique of indefinitely closer approximations to a given magnitude. It depends