CHAPTER XV

On Its Sixth Inexpressible Effect

Folio 0027 (f. 6r) Initial
It is impossible for any rational quantity to be divided according to our said proportion without each of its parts being an irrational line called an apotome (residuo, [XIII.6]).

Numerical Demonstration

For example, let $10$ be the rational quantity that is to be divided according to the proportion having a mean and two extremes. I say that of necessity each of its parts must be an apotome (residuo):

  • One part, namely the lesser, will be $15 - \sqrt{125}$;

  • The other, namely the greater, will be $\sqrt{125} - 5$.

Wherefore it appears that each part is an apotome (residuo)—for so such lines are called in the art according to the seventy-ninth [seventy-third] proposition of the Tenth Book ([X.73]). And this effect we have from the sixth proposition of the Thirteenth Book ([XIII.6]).