Commentary on Proposition X.35

Back to Proposition X.35.

[Folio 0086 | p. 82]

Take the medial straight lines found in x. 32 (2nd part), viz.

$$\rho \lambda^{\frac{1}{4}}, \quad \rho \lambda^{\frac{1}{4}} / \sqrt{1 + k^2}.$$

Solve the equations

$$\left. \begin{aligned} x + y &= \rho \lambda^{\frac{1}{4}} \\ xy &= \frac{\rho^2 \sqrt{\lambda}}{4(1 + k^2)} \end{aligned} \right\} \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots(1),$$

and then put

$$\left. \begin{aligned} u^2 &= \rho \lambda^{\frac{1}{4}} \cdot x \\ v^2 &= \rho \lambda^{\frac{1}{4}} \cdot y \end{aligned} \right\} \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots(2),$$

where $x$, $y$ are the ascertained values of $x$, $y$.

Then $u$, $v$ are straight lines satisfying the given conditions.

Euclid proves this as follows.

($\alpha$) From (1) it follows [x. 18] that $x \mathbin{\smile} y$. Therefore

$$u^2 \mathbin{\smile} v^2,$$

and

$$u \mathbin{\smile} v.$$

($\beta$)

$$u^2 + v^2 = \rho^2 \sqrt{\lambda}, \text{ which is a medial area} \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots(3).$$

($\gamma$)

$$\begin{aligned} uv &= \rho \lambda^{\frac{1}{4}} \cdot \sqrt{xy} \\ &= \frac{1}{2} \frac{\rho^2 \sqrt{\lambda}}{\sqrt{1 + k^2}}, \text{ which is a medial area} \dots\dots\dots\dots\dots\dots(4); \end{aligned}$$

therefore $uv$ is medial.

($\delta$)

$$\rho \lambda^{\frac{1}{4}} \mathbin{\smile} \frac{1}{2} \frac{\rho \lambda^{\frac{1}{4}}}{\sqrt{1 + k^2}},$$

whence

$$\rho^2 \sqrt{\lambda} \mathbin{\smile} \frac{1}{2} \frac{\rho^2 \sqrt{\lambda}}{\sqrt{1 + k^2}}.$$

That is, by (3) and (4),

$$(u^2 + v^2) \mathbin{\smile} uv.$$

The actual values are found thus. Solving the equations (1), we have

$$x = \frac{\rho \lambda^{\frac{1}{4}}}{2}\left(1 + \frac{k}{\sqrt{1 + k^2}}\right),$$

$$y = \frac{\rho \lambda^{\frac{1}{4}}}{2}\left(1 - \frac{k}{\sqrt{1 + k^2}}\right),$$

whence

$$u = \frac{\rho \lambda^{\frac{1}{4}}}{\sqrt{2}} \sqrt{1 + \frac{k}{\sqrt{1 + k^2}}},$$

$$v = \frac{\rho \lambda^{\frac{1}{4}}}{\sqrt{2}} \sqrt{1 - \frac{k}{\sqrt{1 + k^2}}}.$$

[Folio 0087 | p. 83]

According as $\rho$ is of the form $a$ or $\sqrt{A}$, we have a variety of forms for $u$, $v$, arrived at by using the same transformations as in the notes on x. 30 and x. 32 (second part), e.g.

$$(1) \quad \sqrt{\frac{(a + \sqrt{C})\sqrt{B}}{2}}, \quad \sqrt{\frac{(a - \sqrt{C})\sqrt{B}}{2}};$$

$$(2) \quad \sqrt{\frac{(\sqrt{A} + \sqrt{C})\sqrt{B}}{2}}, \quad \sqrt{\frac{(\sqrt{A} - \sqrt{C})\sqrt{B}}{2}};$$

$$(3) \quad \sqrt{\frac{(\sqrt{A} + c)\sqrt{B}}{2}}, \quad \sqrt{\frac{(\sqrt{A} - c)\sqrt{B}}{2}};$$

and the expressions in (2), (3) with $b$ in place of $\sqrt{B}$.