Commentary on Proposition X.90

Back to Proposition X.90.

[Folio 0194 | p. 190]

Let $\rho$ be the given rational straight line.

Take numbers $p$, $(m+n)$, $n$ which have not to one another the ratio of a square number to a square number, $m$, $n$ being also chosen such that the ratio $(m+n) : m$ is not that of square to square.

Take $x$, $y$ such that

$$p : (m+n) = \rho^2 : x^2 \text{ ..............................(1),}$$

$$(m+n) : n = x^2 : y^2 \text{ ..............................(2).}$$

Then shall $(x-y)$ be a sixth apotome.

For, by (1), $x$ is rational and $\sim \rho$ …………………………………..(3).

By (2), since $x$ is rational,

$$y \text{ is rational and } \sim x \text{ .........................................(4).}$$

Thus [(3), (4)] $(x-y)$ is an apotome.

Again, ex aequali,

$$p : n = \rho^2 : y^2\text{,}$$

whence $y \sim \rho$.

Thus $x$, $y$ are both $\sim \rho$.

Lastly, convertendo from (2),

$$(m+n) : m = x^2 : (x^2 - y^2)\text{,}$$

whence $\sqrt{x^2 - y^2} \sim x$.

Therefore $(x-y)$ is a sixth apotome.

From (1) and (2) we have

$$x = \rho \sqrt{\frac{m+n}{p}}\text{,}$$

$$y = \rho \sqrt{\frac{n}{p}}\text{,}$$

so that the sixth apotome may be written

$$\rho \sqrt{\frac{m+n}{p}} - \rho \sqrt{\frac{n}{p}}\text{,}$$

or, more simply,

$$\sqrt{k}\cdot\rho - \sqrt{\lambda}\cdot\rho\text{.}$$

As explained in the note on x. 53, the sixth apotome is the lesser root of the equation

$$x^2 - 2\sqrt{k}\cdot\rho x + (k-\lambda)\rho^2 = 0\text{.}$$