**Theorem 5.9**: The
length of the arc of the
unit circle above the
points* a* and *c *on the *x-*axis is given by

The proof of this theorem requires a Lemma

**Lemma**: Let *P*_{0}, *P*_{1}, . . . , *P*_{n
} be points on the
upper half of the unit
circle where

and

Then *P*_{0}, *P*_{1}, . . . , *P*_{n} is a partition of the
arc on the upper half of the
unit circle if and only if
*x*_{0} < *x*_{1} < . . . < *x*_{n}.