CHAPTER XI¶
On Its Second Essential Effect
If a quantity be divided into two parts, and to one of them be added a quantity such that the square of this sum is quintuple the square of the added quantity, it follows of necessity that the said added quantity is the half of the first quantity divided into those two parts, and that the part to which it was added is its greater part, and that the whole first quantity is divided into those two parts according to our proportion ([XIII.2]).
Numerical Demonstration
For example, let $15 - \sqrt{125}$ and $\sqrt{125} - 5$ be taken as the two integral parts of a quantity [namely $10$], and to one of them—namely $\sqrt{125} - 5$—let $5$ be added as a third quantity. Their sum will be $\sqrt{125}$, whose square is $125$; while the square of the added quantity ($5$) is $25$. Whence $125$ is quintuple to $25$, the square of the added quantity:
$$125 = 5 \times 25$$
I say that $\sqrt{25}$, namely $5$, is the half of the first quantity divided into those two parts; and that the part to which it was added ($\sqrt{125} - 5$) is the greater part of the said first quantity ($10$) divided according to our proportion having a mean and two extremes.
And this is the converse of the preceding effect, just as the second proposition of the Thirteenth Book ([XIII.2]) concludes geometrically.