If an odd number measure an even number, it will also measure the half of it.
For let the odd number A
measure the even number B
; I say that it will also measure the half of it.
For, since A
, let it measure it according to C
; I say that C
is not odd.
For, if possible, let it be so.
Then, since A
according to C
, therefore A
by multiplying C
has made B
is made up of odd numbers the multitude of which is odd.
is odd: [IX. 23
] which is absurd, for by hypothesis it is even.
is not odd; therefore C
an even number of times.
For this reason then it also measures the half of it. Q. E. D.