Commentary on Proposition X.81

Back to Proposition X.81.

[Folio 0176 | p. 172]

As the irrationality of the second apotome of a medial straight line was deduced [x. 75] from the irrationality of an apotome, so the present theorem is reduced to x. 79.

Suppose, if possible, that $(x - y)$, $(x' - y')$ are the same second apotome of a medial straight line; and let (say) $x$ be greater than $x'$.

Apply $(x^2 + y^2)$, $2xy$ and also $(x'^2 + y'^2)$, $2x'y'$ to a rational straight line $\sigma$, i.e. put

$$\left. \begin{aligned} x^2 + y^2 &= \sigma u \\ 2xy &= \sigma v \end{aligned} \right\} \quad\text{and}\quad \left. \begin{aligned} x'^2 + y'^2 &= \sigma u' \\ 2x'y' &= \sigma v' \end{aligned} \right\}.$$

Dealing with $(x - y)$ first, we have: $(x^2 + y^2)$ is a medial area, and $2xy$ is also a medial area. Therefore $u$, $v$ are both rational and $\smile \sigma$ ……………………………………..(1).

$$\begin{aligned} &\text{Also, since } x \underset{-}{\sim} y, & x &\smile y, \\ &\text{so that } & x^2 &\smile xy, \\ &\text{whence, as usual, } & x^2 + y^2 &\smile 2xy, \\ &\text{that is, } & \sigma u &\smile \sigma v, \\ &\text{and therefore } & u &\smile v \text{ .....................................................(2).} \end{aligned}$$

Thus [(1) and (2)] $u$, $v$ are rational and $\underset{-}{\sim}$, so that $(u - v)$ is an apotome. Similarly $(u' - v')$ is proved to be the same apotome. Hence this apotome is formed in two ways: which contradicts x. 79. Therefore the original hypothesis is false, and a second apotome of a medial straight line is uniquely formed.