PROPOSITION 77
If from a straight line there be subtracted a straight line which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial, but twice the rectangle contained by them rational, the remainder is irrational: and let it be called
that which produces with a rational area a medial whole.
For from the straight line
AB let there be subtracted the straight line
BC which is incommensurable in square with
AB and fulfils the given conditions; [
X. 34] I say that the remainder
AC is the irrational straight line aforesaid.
For, since the sum of the squares on
AB,
BC is medial, while twice the rectangle
AB,
BC is rational, therefore the squares on
AB,
BC are incommensurable with twice the rectangle
AB,
BC; therefore the remainder also, the square on
AC, is incommensurable with twice the rectangle
AB,
BC. [
II. 7, X. 16]
And twice the rectangle
AB,
BC is rational; therefore the square on
AC is irrational; therefore
AC is irrational.
And let it be called
that which produces with a rational area a medial whole. Q. E. D.