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 $Q = \tuple {a, 0}$.

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

 

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Historical Note

The length of a logarithmic spiral was first found by Evangelista Torricelli in $1645$.

This was the first time anybody had found the length of a non-straight-line curve for anything other than a circle.

Before this had been done, few people could accept that this was possible to do.

For example, René Descartes had stated in his La Géométrie in $1637$:

Geometry should not include lines that are like strings, in that they are sometimes straight and sometimes curved, since the ratios between straight and curved lines are not known, and I believe cannot be discovered by human minds.

Galileo's response was:

Who is so blind as not to see that, if there are two equal straight lines, one of which is then bent into a curve, that curve will be equal to the straight line?

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.15$: Torricelli ($\text {1608}$ – $\text {1647}$)