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, . . . , P_n be points on the upper half of the unit circle where

and

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

Proof of the Lemma

Proof of Theorem 5.9

next theorem (5.10)