Archimedes estimated the value of $\pi$ by finding the perimeters of regular polygons inscribed in a circle and circumscribed around the circle. He managed to establish that $3\frac{10}{71} < \pi < 3\frac{1}{7}$.

Before he could find the perimeters of polygons he need to be able to calculate square roots. How did he calculate square roots? He didn't have a calculator but needed to work to an appropriate degree of accuracy. To do this he used what we now call numerical roots.

How might he have calculated $\sqrt{3}$?

This must be somewhere between $1$ and $2$. How do I know this? Now calculate the average of $\frac{3}{2}$ and 2 (which is 1.75) - this is a second approximation to $\sqrt 3$. i.e. we are saying that a better approximation to $\sqrt 3$ is $$x_{n+1} = \frac{(\frac{3}{x_n} + x_n)}{2}$$ where $x_n$ is an approximation to $\sqrt 3$ .

We then repeat the process to find the new (third) approximation to $\sqrt{3}$ $$\sqrt{3} \approx {(3 / 1.75 + 1.75) \over {2}} = 1.73214... $$ to find a fourth approximation repeat this process using 1.73214 and so on...

How many approximations do I have to make before I can find $\sqrt{3}$ correct to five decimal places?

Why do you think it works?

Will it always work no matter what I take as my first approximation and does the same apply to finding other roots?