CHAPTER XII

On Its Third Singular Effect

Folio 0027 (f. 6r) Initial
If a quantity be divided according to our proportion, and to its lesser part be added the half of the greater part, the square of their sum will always be quintuple the square of the half of the said greater part ([XIII.3]).

Numerical Demonstration

For example, let $10$ be the quantity divided according to our Divine Proportion, so that one part—namely the greater ($a$)—is $\sqrt{125} - 5$, and the lesser ($b$) is $15 - \sqrt{125}$.

I say that if to $15 - \sqrt{125}$, which is the lesser part, we add the half of $\sqrt{125} - 5$, which is the greater part, the sum of the lesser part and the said half, when multiplied by itself, will be five times the square of the half of the said greater part. And this appears as follows:

  1. The half of the greater part ($\sqrt{125} - 5$) is:

    $$\frac{a}{2} = \sqrt{31\frac{1}{4}} - 2\frac{1}{2}$$

  2. Adding this to $15 - \sqrt{125}$, which is the lesser part ($b$), yields:

    $$b + \frac{a}{2} = (15 - \sqrt{125}) + \left(\sqrt{31\frac{1}{4}} - 2\frac{1}{2}\right) = 12\frac{1}{2} - \sqrt{31\frac{1}{4}}$$

  3. Multiplying $12\frac{1}{2} - \sqrt{31\frac{1}{4}}$ by $12\frac{1}{2} - \sqrt{31\frac{1}{4}}$ produces:

    $$\left(b + \frac{a}{2}\right)^2 = 187\frac{1}{2} - \sqrt{19531\frac{1}{4}}$$

    and this is called the square of the sum.

  4. Next, let the half of the said greater part also be squared—that is, multiply $\sqrt{31\frac{1}{4}} - 2\frac{1}{2}$ by $\sqrt{31\frac{1}{4}} - 2\frac{1}{2}$—which will make:

    $$\left(\frac{a}{2}\right)^2 = 37\frac{1}{2} - \sqrt{781\frac{1}{4}}$$

    and this is called the square of the half of the greater part, which is converterly one-fifth ($\frac{1}{5}$) of the square of the sum:

    $$5 \times \left(37\frac{1}{2} - \sqrt{781\frac{1}{4}}\right) = 187\frac{1}{2} - \sqrt{25 \times 781\frac{1}{4}} = 187\frac{1}{2} - \sqrt{19531\frac{1}{4}}$$

Consequently, the said square of the sum is quintuple the square of the half of the said greater part of $10$ thus divided. And this power is to be highly esteemed along with the others, as is proved geometrically in full by the third proposition of the Thirteenth Book ([XIII.3]) of our author.