Spirals

Summary: §526. The general transcendental-curve family expressed in polar coordinates $z=CM$ (distance from center), $s=\angle ACM$ (angle, measured as the arc on a unit circle). Any equation in $z$ and $s$ defines a spiral. Multi-valued in the same way as the sine-line: $s,s+2\pi ,s+4\pi ,\dots$ and $-s,2\pi -s,\dots$ all represent the same position of the radius $CM$, so the line $CM$ extended cuts the curve in infinitely many points.

Sources: chapter21 §526, figures106-108 (figure 108).

Last updated: 2026-05-12

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Setup (§526)

A spiral is a curve expressed in polar coordinates around a fixed center $C$:

• $z=CM$ — distance from center to a point on the curve.
• $s=\angle ACM$ — angle made by $CM$ with a fixed axis $CA$, measured as the unit-circle arc.

Any equation $F(z,s)=0$ in these variables defines a spiral. The same point can be described by $s+2k\pi$ for any integer $k$, and (on the negated radius) by $-s,2\pi -s,\dots$, so for any radius position there are infinitely many $s$-values to substitute into the equation. Each substitution can give a different $z$, so the straight line $CM$ extended intersects the spiral in infinitely many points (or none, if $z$ becomes complex). This is the polar version of sine-line multi-valued-ness.

The setup figure (§526, figure 108)

Figure 108 shows the generic situation: a point $M$ on a curve $AZ$ at distance $CM=z$ from the center $C$, angle $ACM=s$.

Relation to chapter 17

Chapter 17’s curves-from-polar-coordinates used polar coordinates to describe algebraic curves: equations rational in $\sin s,\cos s$ rationalize to polynomial in $z\sin s=y,z\cos s=x$. The spirals of chapter 21 are precisely the polar curves where the equation in $z,s$ does not rationalize — typically because $s$ appears not just inside $\sin ,\cos$ but as a free variable (linear, polynomial, or under a $\log$).

The four headline examples

Euler catalogues four spirals:

Spiral	Equation	Page
Archimedean	$z=as$	spiral-of-archimedes
Hyperbolic	$z=a/s$	hyperbolic-and-lemniscate-spirals
Lemniscate	$z=a\sqrt{n^2-s^2}$	hyperbolic-and-lemniscate-spirals
Logarithmic	$s=n\log (z/a)$	logarithmic-spiral

The first three are “algebraic in $z,s$” (the equation between $z$ and $s$ is polynomial), but transcendental as curves because $s$ is fundamentally a transcendental function of the rectangular coordinates. The fourth is transcendental in the polar equation itself.

Place in the chapter

The spirals close chapter 21 as a fifth, distinct cluster of transcendental curves, characterized by polar formulation. Euler notes that “transcendental equations in $z,s$ could vastly extend this discussion” — but he singles out only the logarithmic spiral from that broader family, due to its equiangular property.

Figures

Figures 106–108

Related pages

• chapter-21-on-transcendental-curves
• spiral-of-archimedes — simplest spiral.
• hyperbolic-and-lemniscate-spirals — variants in $1/s$ and $\sqrt{n^2-s^2}$.
• logarithmic-spiral — equiangular spiral, transcendental in equation.
• curves-from-polar-coordinates — chapter 17 polar setup, here reused.
• transcendental-curves