Logarithmic Spiral

Summary: §528. The polar curve $s=n\log (z/a)$, equivalently $z=a{e}^{s/n}$. The single transcendental-equation spiral Euler singles out. Equiangular property: every radius from the center meets the curve at the same angle, $\arctan n$. For $n=1$ the angle is $\pi /4$ (the semi-rectangular logarithmic spiral). Drawn as figure 111.

Sources: chapter21 §528, figures111-114 (figure 111).

Last updated: 2026-05-12

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The equation (§528)

$s=n\log \frac{z}{a},\text{equivalently}z=a{e}^{s/n}.$

The polar angle $s$ is proportional to the logarithm of the radius. Equivalently, the radius $z$ grows exponentially with the angle — geometric progression of radii for arithmetic progression of angles, exactly mirroring the logarithmic-curve’s rectangular property.

The angle $s$ is proportional to the logarithm of the distance $z$. For this reason the curve is called a logarithmic spiral and because of many important properties it is especially noteworthy. (source: chapter21, §528)

This is the only curve in §§526–528 whose polar equation is itself transcendental (the others are algebraic in $z,s$, transcendental only because $s$ is). Euler singles it out for its uniqueness.

Equiangular property (§528, figure 111)

The principal property of this curve is that any straight line from the center $C$ intersects the curve in equal angles. (source: chapter21, §528)

Derivation. Take a radius $CM=z$ at angle $s$, and a nearby radius $CN$ at angle $s+v$. Then $CN=a{e}^{(s+v)/n}=z{e}^{v/n}$. Draw the circular arc $ML$ from $M$ to the new radius, of length $zv$ (since arc = radius × angle in unit-circle convention). The remaining segment from $L$ to $N$ along the radius:

$LN=CN-CM=z(e^{v/n}-1)=z\left(\frac{v}{n}+\frac{v^2}{2n^2}+\frac{v^3}{6n^3}+\cdots \right).$

Now ML/LN = \tan(\text{angle between curve and radius at L, in the limit}):

$\frac{ML}{LN}=\frac{zv}{z(v/n+v^2/(2n^2)+\cdots )}=\frac{n}{1+v/(2n)+v^2/(6n^2)+\cdots }.$

As $v\to 0$ (so $L$ converges to $M$ along the curve), the ratio tends to $n$:

$\boxed{\tan (\text{angle between radius and curve})=n\text{everywhere}.}$

The angle is constant, independent of $s$ or $z$. This is the equiangular (or isogonal) property.

Semi-rectangular case (§528)

If $n=1$, the angle will always be half of a right angle. For this reason such a curve is called a semi-rectangular logarithmic spiral. (source: chapter21, §528)

$\arctan 1=\pi /4$, hence the name. This particular spiral is closely related to the curve of similar curves and to self-similar growth in nature (nautilus shell, hurricane arms, galaxy spirals).

Figure 111 features

The figure shows the spiral coiling outward from the center $C$, indefinitely. The radii $CM,C{A}^1,C{B}^1,\dots$ all make the same constant angle with the curve at their intersection points. $A,A^1,A^{-1},B,B^1,\dots$ mark points where the curve crosses two perpendicular reference axes through $C$.

Why “logarithmic”

The name comes from the form $s=n\log (z/a)$: the polar angle is a logarithm of the radius. Some sources call it the equiangular spiral for the property, or the Bernoulli spiral — Jacob Bernoulli was so taken with its self-similarity under inversion that he asked for one to be carved on his tombstone (eadem mutata resurgo — “though changed I rise again the same”).

Place in the chapter

The logarithmic spiral is the closing example of Euler’s transcendental-curve catalogue. Picked out for two reasons:

1. It is the only spiral in §§526–528 whose polar equation is itself transcendental (the others $z=as,z=a/s,z=a\sqrt{n^2-s^2}$ are algebraic in $z,s$).
2. Its equiangular property — pre-calculus derivation included in §528 — is “especially noteworthy” and prefigures the differential-geometric notion of constant geodesic curvature.

Figures

Figures 111–114

Related pages

• chapter-21-on-transcendental-curves
• spirals — general setup.
• spiral-of-archimedes — linear sibling.
• hyperbolic-and-lemniscate-spirals — algebraic-in-$z,s$ siblings.
• logarithmic-curve — rectangular cousin (same geometric–arithmetic correspondence in different coordinate frame).
• similar-curves — chapter 18 framework, here realized as self-similarity under dilation about $C$.