PROPOSITION 108.
If from a rational area a medial area be subtracted, the “side”
of the remaining area becomes one of two irrational straight lines, either an apotome or a minor straight line.
For from the rational area
BC let the medial area
BD be subtracted; I say that the “side”
of the remainder
EC becomes one of two irrational straight lines, either an apotome or a minor straight line.
For let a rational straight line
FG be set out, to
FG let there be applied the rectangular parallelogram
GH equal to
BC, and let
GK equal to
DB be subtracted; therefore the remainder
EC is equal to
LH.
Since then
BC is rational, and
BD medial, while
BC is equal to
GH, and
BD to
GK, therefore
GH is rational, and
GK medial.
And they are applied to the rational straight line
FG; therefore
FH is rational and commensurable in length with
FG, [
X. 20] while
FK is rational and incommensurable in length with
FG; [
X. 22] therefore
FH is incommensurable in length with
FK. [
X. 13]
Therefore
FH,
FK are rational straight lines commensurable in square only; therefore
KH is an apotome [
X. 73], and
KF the annex to it.
Now the square on
HF is greater than the square on
FK by the square on a straight line either commensurable with
HF or not commensurable.
First, let the square on it be greater by the square on a straight line commensurable with it.
Now the whole
HF is commensurable in length with the rational straight line
FG set out; therefore
KH is a first apotome. [
X. Deff. III. 1]
