CHAPTER XXII

On Its Thirteenth Most Worthy Effect

[Folio 0030 (f. 7v)]

For its thirteenth effect, it is no small matter of wonder that without its aid one can never construct the equilateral and equiangular pentagon—that is, the figure of five equal sides adduced above in the Ninth Effect and to be adduced again below—and that without the said pentagon, as will be told, it is impossible either to form or to imagine the noblest body above all the others, called the Dodecahedron (duodecedron)—that is, the body of twelve equilateral and equiangular pentagons, otherwise called the body of twelve pentagonal bases—whose form, as will be told, the divine Plato [in his Timaeus] attributed for the most fitting reasons to the Fifth Essence, namely to Heaven ([IV.10], [IV.11], [XIII.17]).

Euclid’s Deductive Architecture in Books II, IV, V, and VI

To this end, our Philosopher in the tenth proposition of the Fourth Book ([IV.10]) teaches us how to construct an isosceles triangle of this condition: namely, that each of the two angles at its base is double the remaining angle [at the apex, i.e., $72^\circ, 72^\circ, 36^\circ$].

And he did this because, when we wish to know how to form the equilateral and equiangular pentagon, and to inscribe and circumscribe it in and about a circle—that is, to form it both inside and outside a given circle with exactness—it would have been impossible unless he had first taught us how to construct the said triangle, as appears from the eleventh and twelfth propositions of the said Fourth Book ([IV.11], [IV.12]).

Now, in order to construct the said triangle, it is of necessity required to divide a line according to our Divine Proportion, as he shows us in the said tenth proposition of the Fourth Book ([IV.10]).

It is true that in that place ([IV.10]) Euclid does not state in words that the said line is divided “according to the said proportion and its conditions,” because he had not yet given us knowledge of what a proportion is—which topic he reserves for his Fifth Book! For it is never his custom to introduce into his demonstrations subsequent doctrines of which knowledge has not yet been given; rather, he uses only antecedent propositions, and this rigorous deductive order is maintained throughout all fifteen of his Books.

Therefore, for the purpose of the said triangle in [IV.10], he does not say “divide the said line according to the proportion having a mean and two extremes,” but says instead: “by the eleventh proposition of the Second Book (:just:`II.11`), divide it into two such parts that the square of one part is equal to the product [rectangle] of the other part and the whole line”—which in substance and power means nothing other than dividing it according to our Divine Proportion, as appears from the third definition of the Sixth Book ([VI.def.3]) and the twenty-ninth [thirtieth] proposition of the same Book ([VI.30]), and as we ourselves explained above when we clarified how its mean and its extremes are to be understood under the First Effect [Chapter VII].