CHAPTER X¶
Sequel to the First Proposed Effect
Once these things have been well noted, let us return to its first proposed effect [Chapter VII] and make it clear with evident examples. For its elucidation, let us resume the very same case of $10$ adduced above, without troubling ourselves with other laborious quantities, since the very same property that is stated here occurs invariably in every quantity whatever. And for the fuller understanding of Your Highness, we shall pursue all the remaining effects by way of Arithmetic, while presupposing throughout that the rigorous scientific proofs of everything contained in our treatise are assigned geometrically with all care by our philosopher Euclid in the propositions we shall cite, according to the requirements of the conclusions.
Arithmetical Demonstration of the First Effect ([XIII.1])
I say, therefore, that when $10$ is divided according to our proportion, its greater part is $\sqrt{125} - 5$. If to this greater part, in accordance with the said First Effect, we add $5$—that is, the half of the whole $10$—their sum will make $\sqrt{125}$ exactly; because that $- 5$ (meno 5) is restored and filled up (se vene a restorare e a rempire) by the $+ 5$ (più 5), the half of $10$:
$$(\sqrt{125} - 5) + 5 = \sqrt{125}$$
When this sum, namely $\sqrt{125}$, is multiplied by itself, it makes $125$ for its square. And this $125$ is five times the square of the half of $10$, which half is $5$ and its square is $25$:
$$(\sqrt{125})^2 = 125 = 5 \times 25 = 5 \times 5^2$$
Whence $125$ is exactly quintuple to the said $25$, the square of the said half of $10$, just as was stated.