165. Al-Karhi's rule for approximating the square root. If $a^{2}$ is the greatest square contained in a given number $N$, and $r$ is the remainder, then

$$
\sqrt{N}=\sqrt{a^{2}+r}=a+\frac{r}{2 a+1}, \text{ if } r<2 a+1
$$

Explain how Al-Karhi might have derived this rule. Evaluate the error by calculating $\sqrt{415}$ in the usual way to an accuracy of $0.001$.
165. If \(a^{2}\) is the greatest square contained in a given number \(N\), then let \(\sqrt{N}=\sqrt{a^{2}+r}=a+x^{2}\), where \(x<1\). Then \(a^{2}+r=a^{2}+2 a x+x^{2}\), or \(r=2 a x+x^{2}=x(2 a+x)\), from which

\[
x=\frac{r}{2 a+x}
\]

$8^{*}$

Al-Karhi replaces \(x\) in the denominator with 1 and obtains:

\[
\begin{gathered}
x=\frac{r}{2 a+1}<\frac{r}{2 a+x} \\
\sqrt{415}=20.371
\end{gathered}
\]

according to Al-Karhi:

\[
V \overline{415}=20+\frac{15}{2 \cdot 20+1}=20+\frac{15}{41}=20.366
\]