Length of Arc of Cycloid/Proof 2

Proof
Consider the tangent $PQ$ to both the generating circle and the cycloid itself.


 * ArcLengthOfCycloid.png

By :
 * $PQ = 2 PR$

In the limit, where $P$ is at the cusp, the tangent $PQ$ is perpendicular to the straight line along which the generating circle rolls.

At this point:
 * $PQ = 2 a$.

Thus at this point:
 * $PR = 4 a$

But $4 a$ is half the length of one arc of $C$.

Hence the result.

Historical Note
This result was demonstrated by in $1658$.