CHAPTER XVIII¶
On Its Ninth Effect, Surpassing All the Others
Geometric and Numerical Demonstration
For example, let the equilateral pentagon be abcde, and from the endpoints c and a let the chord ac be drawn subtending the angle b; and from the endpoints b and e let the other chord be be drawn subtending the angle a.
I say that these two lines ac and be divide one another at the point f according to the proportion having a mean and two extremes ([XIII.8]), and the greater part of each is exactly equal to the side of the said pentagon:
Of the line ac, the greater part is cf, and the lesser part is af;
Of the line be, the greater part is ef, and the lesser part is bf;
And each of these greater parts (cf and ef) is always equal to the side of the said pentagon (ab $= $ bc $= $ cd $= $ de $= $ ea).
By mathematicians these two lines (ac and be) are called by another name the chords of the pentagonal angle (corde del’angolo pentagonico).
Thus, if each of the said chords (ac and be) were $10$ (for they are equal when their pentagon in the circle is equilateral and equiangular):
cf would be $\sqrt{125} - 5$, and af would be $15 - \sqrt{125}$;
The part ef would similarly be $\sqrt{125} - 5$, and bf would be $15 - \sqrt{125}$;
And the side of the pentagon itself would likewise be $\sqrt{125} - 5$!