Commentary on Proposition X.80

Back to Proposition X.80.

[Folio 0173 | p. 169]

Suppose, if possible, that the same first apotome of a medial straight line can be expressed in terms of the required character in two ways, so that

$$x - y = x' - y',$$

and suppose that $x > x'$.

In this case $x^2 + y^2$, $(x'^2 + y'^2)$ are both medial areas, and $2xy$, $2x'y'$ are both rational areas; and

$$x^2 + y^2 - (x'^2 + y'^2) = 2xy - 2x'y'.$$

Hence x. 26 is contradicted again; therefore etc.