arccos-log-curve

The Curve $y=\arccos \log z$

Summary: §525. The §511 curve $y=\cos \log x$ inverted, written as $\arccos y=\log z$ or $z=e^{\arccos y}$. Combines logarithms and circular arcs. The axis is met at $AD=e^{\pi /2},A{D}^1=e^{3\pi /2},\dots$ (geometric progression). The curve oscillates and is tangent to two axis-parallel lines at points $E,F$ also in geometric progression. Asymptote is the finite interval $BC$, not an infinite line — a non-algebraic phenomenon. Drawn as figure 107.

Sources: chapter21 §525, figures106-108 (figure 107).

Last updated: 2026-05-12

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Setup (§525)

The §511 complex-exponent curve $2y=x^i+x^{-i}$ collapses to $y=\cos \log x$. Solving for $\log x$ gives the inverted form:

$y=\cos \log z⟺\arccos y=\log z⟺z=e^{\arccos y}.$

Now $z$ is the abscissa, $y$ the ordinate. Negative $z$ is excluded (no continuous part there).

Geometric features (§525, figure 107)

Reading figure 107 (axis vertical, labeled $BC$, with $A$ inside the page):

Axis intersections ($y=0\Rightarrow \arccos 0=(2n+1)\pi /2$):

$AD=e^{\pi /2},A{D}^1=e^{3\pi /2},A{D}^2=e^{5\pi /2},A{D}^3=e^{7\pi /2},\dots$

A geometric progression with ratio $e^{\pi }$.

Inward intersections ($\arccos 0=-(2n+1)\pi /2$):

$A{D}^{-1}=e^{-\pi /2},A{D}^{-2}=e^{-3\pi /2},\dots$

Converging to $A$ as $n\to \infty$. These are also a geometric progression, with the same ratio $e^{\pi }$.

Tangencies. The curve is tangent to two lines parallel to the axis through $A$ and $B$ (at distances $AB=AC=1$ — i.e. where $y=\pm 1$, $\arccos$ reaches its extremes $0$ and $\pi$). The points of tangency $E,F$ are stacked at distances $e^0,e^{\pi },e^{2\pi },e^{3\pi },\dots$ from $A$ — yet again a geometric progression.

The remarkable asymptote. As $y$ oscillates and the curve loses amplitude (it doesn’t, actually — the $\cos$ stays bounded between $\pm 1$, but the spacing of axis crossings grows geometrically), the inflections approach the interval $BC$ at distance $\pm 1$ from the axis:

Furthermore, the infinite number of inflections of the curve approach the line $BC$ and finally they move into that line. A particular characteristic of this curve is that the asymptote is not an infinite straight line, but the finite interval $BC$. This property of the curve keeps it from being an algebraic curve. (source: chapter21, §525)

The “asymptote” is a finite segment — a striking phenomenon that no algebraic curve can exhibit (where asymptotes are full lines or curves).

Why it qualifies as transcendental of mixed genus

The equation $z=e^{\arccos y}$ requires both:

• $\exp$ (or $\log$) — for the relation between $z$ and the arc.
• $\arccos$ — for the relation between $y$ and the arc.

So the curve sits at the intersection of the logarithmic genus (chapter 21 second cluster) and the circular-arc genus (third cluster). Euler treats it as the natural close of the catalogue before turning to spirals.

Place in the chapter

The §525 curve is the only mixed-genus example Euler presents, and it is included specifically because the §511 derivation already showed how complex exponents produce real curves involving both logarithms and arcs. The finite-asymptote phenomenon is its signature.

Figures

Figures 106–108

Related pages

• chapter-21-on-transcendental-curves
• transcendental-curves — §511 motivation: $2y=x^i+x^{-i}=2\cos \log x$.
• logarithmic-curve — logarithmic-genus parent.
• sine-line, cycloid — circular-arc-genus relatives.
• transcendental-curves