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   P0, P1, . . . , Pn   be points on the upper half of the unit circle where

Pi = (xi, yi)

and

a = xo < xn = c

Then   P0, P1, . . . , Pn   is a partition of the arc on the upper half of the unit circle if and only if   x0 < x1 < . . . < xn.

Proof of the Lemma

Proof of Theorem 5.9

next theorem (5.10)