Commentary on Proposition X.79

Back to Proposition X.79.

[Folio 0172 | p. 168]

This proposition proves the equivalent of the well-known theorem of surds that, if $a - \sqrt b = x - \sqrt y$, then $a = x, b = y$; and, if $\sqrt a - \sqrt b = \sqrt x - \sqrt y$, then $a = x, b = y$.

The method of proof corresponds to that of x. 42 for positive signs.

Suppose, if possible, that an apotome can be expressed as $(x - y)$ and also as $(x' - y')$, where $x, y$ are rational straight lines commensurable in square only, and $x', y'$ are so also.

Of $x, x'$, let $x$ be the greater.

Now, since

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

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

But $(x^2 + y^2), (x'^2 + y'^2)$ are both rational, so that their difference is a rational area.

On the other hand, $2xy, 2x'y'$ are both medial areas, being of the form $\sqrt k \cdot \rho^2$; therefore the difference between two medial areas is rational: which is impossible [x. 26].

Therefore etc.