# Length of Logarithmic Spiral

Jump to navigation
Jump to search

## 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

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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 ($\text {1608}$ – $\text {1647}$)