404
EUCLID
The Book begins with the famous proposition, on which the
‘ method of exhaustion ’ as used in Book XII depends, to the
effect that, if from any magnitude there be subtracted more
than its half (or its half simply), from the remainder more than
its half (or its half), and so on continually, there will at length
remain a magnitude less than any assigned magnitude of the
same kind. Proposition 2 uses the process for finding the
G. C. M. of two magnitudes as a test of their commensurability
or incommensurability: they are incommensurable if the process
never comes to an end, i.e. if no remainder ever measures the
preceding divisor ; and Propositions 3, 4 apply to commen
surable magnitudes the method of finding the G. C. M. of two
or three numbers as employed in VII, 2, 3. Propositions 5
to 8 show that two magnitudes are commensurable or incom
mensurable according as they have or have not to one another
the ratio of one number to another, and lead up to the funda
mental proposition (9) of Theaetetus already quoted, namely
that the sides of squares are commensurable *or incommen
surable in length according as the squares have or have not to
one another the ratio of a square number to a square number,
and conversely. Propositions 11-16 are easy inferences as to
the commensurability or incommensurability of magnitudes
from the known relations of others connected with them ;
e.g. Proposition 14 proves that, if a:b = c:d, then, according
as U (a 2 — b 2 ) is commensurable or incommensurable with a,
V(c 2 —d 2 ) is commensurable or incommensurable with c.
Following on this, Propositions 17, 18 prove that the roots of
the quadratic equation ax — x 2 = b 2 /4 are commensurable or
incommensurable with a according as V{a 2 — b 2 ) is commen
surable or incommensurable with a. Propositions 19-21 deal
with rational and irrational rectangles, the former being
contained by straight lines commensurable in length, whereas
rectangles contained by straight lines commensurable in square
only are irrational. The side of a square equal to a rectangle
of the latter kind is called medial ; this is the first in Euclid’s
classification of irrationals. As the sides of the rectangle may
be expressed as p, pVh, where p is a rational straight line,
the medial is Adp. Propositions 23-8 relate to medial straight
lines and rectangles ; two medial straight lines may be either
commensurable in length or commensurable in square only :