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 :