Length of Logarithmic Spiral

Theorem
Consider a logarithmic spiral $S$ given by the equation:
 * $r = a e^{b \theta}$

Construct a tangent to $S$ at the point $Q = \left({a, 0}\right)$.

Let the tangent cross the $y$-axis at $P$.

Then the length of $PQ$ equals the total length of $S$ from $P$ inwards to the origin.

Proof

 * LogarithmicSpiralLength.png

Historical Note
This result was demonstrated by in $1645$.