CHAPTER XX¶
On Its Eleventh Most Excellent Effect
If the side of an equilateral hexagon be divided according to our Divine Proportion, its greater part will of necessity always be the side of the regular decagon circumscribed by the very same circle as the hexagon ([XIV.3] [Campanus corollary to [XIII.9]]).
Numerical Demonstration
For example, if the side of the equilateral hexagon be $10$, and it be divided in the manner stated, its greater part will be $\sqrt{125} - 5$, which I say is the exact side of the decagon inscribed in the very same circle, whose diameter will be $20$ ([IV.15]). And this is concluded by the third proposition of the Fourteenth Book (Campanus XIV.3 / [XIII.9]).
Whence, manifestly, once the side of either polygon is known, the side of the other is easily found; and likewise, given the diameter of the circle, or its circumference, or its area, or any part thereof, we can always arrive through it at the knowledge of either the hexagon or the decagon from the other, and conversely in every way for the circle, hexagon, decagon, and also the equilateral triangle—operating arithmetically and geometrically, which is a most useful thing, just as was stated above in the Ninth Effect [Chapter XVIII] concerning the pentagon.
Marginal Woodcut Diagrams (Paganini 1509)¶
Folio 0029 Diagram 1¶