Archimedes would have liked the following function in two real variables
Can you think of a reason why, given the following expressions?
Here’s the pattern. When the point lies on the unit circle, say in the first quadrant, then is a pretty good approximation of the arc length from to this point. I like the first example above where a quarter circle already gives a very decent approximation to . The last example above estimates with one twelfth of a circle. The result is close enough to have the famous approximation
in its partial fraction expansion. Where did this formula come from? The idea was to choose parameters a, b, c such that
The choices 14, 6 and 9 give the following approximation
These are also optimal in the sense that all other choices approximate to a lower order. Archimedes started with one sixth of a circle and halved the arc in each subsequent step. With this order seven approximation the error in the approximation of is reduced by a factor of 64 in each step. Using the function instead of chord lengths (as Archimedes did) requires only a third of the number of steps for the same accuracy.