Commentary on Proposition X.51

Back to Proposition X.51.

[Folio 0114 | p. 110]

Take numbers $m$, $n$ such that $(m + n)$ has not to either $m$ or $n$ the ratio of square to square.

Take $x$ such that

$$(m + n) : m = k^2 p^2 : x^2,$$

whence

$$x = kp \sqrt{\frac{m}{m + n}} = \frac{kp}{\sqrt{1 + \lambda}}\text{, say.}$$

Then $kp + x$, or $kp + \frac{kp}{\sqrt{1 + \lambda}}$, is a fourth binomial straight line.

For $\sqrt{k^2 p^2 - x^2}$ is incommensurable in length with $kp$, and $kp$ is commensurable in length with $p$.

The corresponding fourth apotome [x. 88] is

$$kp - \frac{kp}{\sqrt{1 + \lambda}}.$$

The equation of which the two expressions are the roots is

$$x^2 - 2kp \cdot x + \frac{\lambda}{1 + \lambda} k^2 p^2 = 0,$$

or

$$x^2 - 2ax + \frac{\lambda}{1 + \lambda} a^2 = 0,$$

where

$$a = kp.$$