Method of Exhaustion


Eudoxus, Greek mathematician 408-355 BC, developed the idea of seeking mathematical solutions using he $\textit{Method on Exhaustion}$: at this stage, you achieve a more accurate figure. To find an increasingly accurate solution to $\sqrt{2}$, for example he produced a ladder of numbers.

Starting with $1$ and $1$ as the first row, he then added those numbers together to start the second row – so $1+1=2$. He then added this number the one above it to get the second number in the second row: $2+1=3$. And so on. As the ladder builds, the ratio of each pair of numbers grows ever closer to the value for $\sqrt{2}$.

$$
\begin{array}{|r|r|l|} \hline
1 & 1 & 1 \div 1 = 1 \\ \hline
2 & 3 & 3 \div 2 = 1.5 \\ \hline
5 & 7 & 7 \div 5 = 1.4 \\ \hline
12 & 17 & 17 \div 12 = 1.416666 \cdots \\ \hline
29 & 41 & 41 \div 29 = 1.4137931 \cdots \\ \hline
70 & 99 & 99 \div 70 = 1.4142857 \cdots \\ \hline
\end{array} \\
\sqrt{2} = 1.4142135 \cdots
$$

An unusual property of this sequence is that if you double the square of the first number it is always 1 more or 1 less than the square of the second number, so $5 \times 5 \times 2 = 50$, while $7 \times 7 = 49$.