CHAPTER XIX¶
On Its Tenth Supreme Effect
Geometric and Numerical Demonstration
For example, let there be two lines thus divided: namely, one line ab divided at c whose greater part is ac, and another line de divided at f whose greater part is df. And what we say of these two lines, we understand likewise of infinitely many others, which can easily be assigned by way of Arithmetic:
Setting ab $= 10$, its greater part ac will be $\sqrt{125} - 5$, and its lesser part cb will be $15 - \sqrt{125}$;
Setting de $= 12$, its greater part df will be $\sqrt{180} - 6$, and its lesser part fe will be $18 - \sqrt{180}$.
I say that everything that can ever happen to one of the said lines when compared, multiplied, divided, or manipulated in any other way whatever, happens identically to the other:
The ratio of each whole line to its greater part is the very same (ab $:$ ac $=$ de $:$ df);
The ratio of each whole line to its lesser part is the very same (ab $:$ cb $=$ de $:$ fe);
And conversely, from each of their parts to the wholes, and likewise in the product of one into its parts and conversely, and in division and subtraction.
Thus the ratio of $10$ to its greater part $\sqrt{125} - 5$ is the exact same as the ratio of $12$ to its greater part $\sqrt{180} - 6$; and the ratio of the sum of $10$ and $\sqrt{125} - 5$ to $\sqrt{125} - 5$ is the exact same as the ratio of the sum of $12$ and $\sqrt{180} - 6$ to $\sqrt{180} - 6$.