Commentary on Proposition X.52

Back to Proposition X.52.

[Folio 0115 | p. 111]

If $m$, $n$ be numbers of the kind taken in the last proposition, take $x$ such that

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

[Folio 0116 | p. 112]

In this case

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

and $x > kp$.

Then $kp \sqrt{1 + \lambda} + kp$ is a fifth binomial straight line.

For $\sqrt{x^2 - k^2 p^2}$, or $\sqrt{\lambda}\cdot kp$, is incommensurable in length with $kp \sqrt{1 + \lambda}$, or $x$; and $kp$, but not $kp \sqrt{1 + \lambda}$, is commensurable in length with $p$.

The corresponding fifth apotome [x. 89] is

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

The equation of which the fifth binomial and the fifth apotome are the roots is

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

or

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

where

$$a = kp \sqrt{1 + \lambda}.$$