Commentary on Proposition X.16

Back to Proposition X.16.

[Folio 0045 | p. 41]

If $a$ be the given straight line, and $x$ the side of the square by which the applied rectangle is to be deficient, the rectangle is equal to $ax - x^2$, which is of course equal to $x(a - x)$. The rectangle may be written $xy$, where $x + y = a$. Given the area $x(a - x)$, or $xy$ (where $x + y = a$), two different applications will give rectangles equal to this area, the sides of the defect being $x$ or $a - x$ ($x$ or $y$) respectively; but the second mode of expression shows that the rectangles do not differ in form but only in position.