CHAPTER XVI¶
On Its Seventh Inestimable Effect
If the side of an equilateral hexagon be joined in a single line to the side of an equilateral decagon, both being understood as inscribed in one and the same circle, their sum will always be a quantity divided according to our said proportion, and its greater part will be the side of the hexagon ([XIII.9]).
Numerical Demonstration
For example, let the side of an equilateral hexagon inscribed in a circle be $\sqrt{125} - 5$, and let the side of the equilateral decagon in the very same circle be $15 - \sqrt{125}$. (The diameter of this circle, which is double the side of the hexagon by [IV.15], will be $2(\sqrt{125} - 5) = \sqrt{500} - 10$.)
I say that the sum of $\sqrt{125} - 5$ and $15 - \sqrt{125}$, which equals $10$, is divided according to our proportion, and that its greater part is $\sqrt{125} - 5$ [the hexagon side] and its lesser part is $15 - \sqrt{125}$ [the decagon side], just as we have repeatedly shown in the division of $10$.
And this is manifest geometrically by the ninth proposition of the Thirteenth Book ([XIII.9]).