Chapter 21 — On Transcendental Curves

Summary: Euler’s catalogue of the principal transcendental curves. After 20 chapters spent entirely on algebraic curves, he turns to the curves whose defining equation cannot be brought to polynomial form. Five clusters: definition and intercendental/complex-exponent curves (§§506–511); the logarithmic curve with its paradoxes (§§512–517); exponential curves (§§518–519); the circular-arc curves — sine line, tangent line, cycloid, epicycloid, $y=\arccos \log z$ — (§§520–525); spirals (§§526–528).

Sources: chapter21, figures99-102, figures103-105, figures106-108, figures109-110, figures111-114.

Last updated: 2026-05-12

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Five strands

1. Definition (§§506–508) and exotic exponents (§§509–511)

A curve is transcendental when the relation between $x$ and $y$ cannot be written as a polynomial equation; equivalently, when $y$ is a transcendental function of $x$ (§506). The set of transcendental functions is much larger than $\{\log ,\arcsin ,\arccos ,\arctan \}$; analysis of the infinite produces innumerable others (§507). A non-irrational algebraic equation has only integer exponents; if some exponent in the defining equation is neither integer nor a fraction, the curve cannot be made non-irrational and is therefore transcendental (§508).

This already produces a simplest sub-genus that Leibniz called intercendental-curves: $y=x^{\sqrt{2}}$ and its kin (§§509–510). Approximating $\sqrt{2}$ by $3/2,7/5,17/12,41/29,99/70,\dots$ gives algebraic curves of order $3,7,17,41,\dots$ — so the true curve has infinite order and is decisively not algebraic. To construct it exactly requires logarithms ($\log y=\sqrt{2}\log x$).

Complex exponents (§511) produce another exotic family: $2y=x^i+x^{-i}$ collapses to $y=\cos \log x$ (using $e^{vi}+e^{-vi}=2\cos v$ from Book I §138). Real-valued despite the imaginary exponents — and used again in §525 as the arccos-log-curve.

2. Logarithmic curve and its paradoxes (§§512–517)

The logarithmic-curve $y=a{e}^{x/b}$ (§§512–514, figure 101) is the canonical first transcendental: arithmetic progression in $x$ produces geometric progression in $y$. Its hallmark is constant subtangent $PT=b$ everywhere — the logarithmic parameter $b$ realized geometrically. The axis is an asymptote in the negative-$x$ direction.

But the curve raises two paradoxes that “never occur in algebraic curves”:

• discrete-points-below-asymptote (§§515, 517). When $x/b$ has an even denominator, $y$ acquires a second negative value. Hence below the asymptote sit infinitely many discrete points, dense in pairs of axis-parallel lines but forming no continuous branch. The same anomaly appears starkly in $y=(-1{)}^x$ (§517).
• infinite-logarithms-paradox (§516). Since $\log 1=0$ but also $\log (-1{)}^2=0$, half of $a$ is both $a/2$ and $a/2+\log (-1)$. Every number has infinitely many logarithms, only one of which is real. Algebraic root extraction has $n$ choices; logarithms have $\infty$.

3. Exponential curves (§§518–519)

Any curve whose equation can be put in a form involving a logarithm is assigned to the logarithmic genus — including the exponentials $y=x^x$ (§518, figure 102). On the positive axis the curve has a minimum at $x=1/e\approx 0.368$ with $y\approx 0.692$; for $x<0$ only discrete points exist (same parity anomaly as the logarithmic curve). The curve “terminates abruptly” at $B$ at the $y$-axis crossing, which violates the algebraic law of continuity but is unavoidable for transcendentals.

The implicit-exponential-curve $z^y=y^z$ (§519, figure 103) is harder. The straight line $y=z$ is one component (since $z^z=z^z$), but the equation is also satisfied by an additional branch $RS$ asymptotic to the axes, parametrized by $z=(1+1/u{)}^u,y=(1+1/u{)}^{u+1}$. The two components meet at $z=y=e$. Rational solution pairs $(2,4),(9/4,27/8),(64/27,256/81),\dots$ are dense on the branch.

4. Circular-arc and combined curves (§§520–525)

The sine-line $y=a\arcsin (x/c)$ (§520, figure 104) is the canonical curve requiring circular arcs. Any vertical line cuts it in infinitely many points $s,\pi -s,2\pi +s,\dots$ — the same multi-valued-ness as the logarithmic curve’s discrete points, but now forming a true continuous oscillation. Two diameters $BC$ and parallels; periods $2a\pi$ apart. Leibniz’s name. The same curve is the cosine line with $y$ shifted by $\pi a/2$.

The tangent-and-secant-lines (§521) are $y=\arctan x$ (infinitely many parallel asymptotes) and $y=\mathrm{arcsec}x$ (infinitely many infinite branches).

The cycloid (§§521–522, figure 105) — path of a point on a circle rolling along a line — has equation $y=\sqrt{2bz-z^2}+a\arcsin (\sqrt{2bz-z^2}/b)$, or in centered form $y=\sqrt{b^2-t^2}+a\arccos (t/b)$. Ordinary when $b=a$ (point on the circumference); curtate / prolate when $b>a$ or $b<a$ (point inside / outside the rolling circle).

epicycloid-and-hypocycloid (§§523–524, figure 106). A circle of radius $a$ rolling on the outside (or inside, $a<0$) of a stationary circle of radius $c$ traces $z=(a+c)\cos (s/c)+b\cos ((a+c)s/(ac))$, $y=(a+c)\sin (s/c)+b\sin ((a+c)s/(ac))$. Algebraic when $(a+c)/a$ is rational; otherwise transcendental. The special case $b=-(a+c)$ degenerates to the center of the stationary circle; expanding $c=2a$ gives a sextic $(z^2+y^2{)}^3-2f^2(z^2+y^2{)}^2+f^4{z}^2=0$.

The arccos-log-curve $y=\arccos \log z$ (§525, figure 107) is the §511 curve $y=\cos \log x$ inverted, expressed as $z=e^{\arccos y}$. Combines both logarithms and arcs. Axis intersected at $AD=e^{\pi /2},A{D}^1=e^{3\pi /2},\dots$ (geometric progression); tangent to two axis-parallel lines at points $E,F$ forming another geometric progression; finite asymptote $BC$ — a non-algebraic phenomenon.

5. Spirals (§§526–528)

Polar equations $z=f(s)$ where $z=CM$ is the distance from a fixed center $C$ and $s$ measures the angle $\angle ACM$ along a unit circle.

• spiral-of-archimedes $z=as$ (§526, figure 109). Tangent at $C$ along $AC$; intersects $AC$ and $BC$ at right angles infinitely many times; diameter $BC{B}^{-1}$.
• Hyperbolic spiral $z=a/s$ (§527). After infinitely many turns approaches line $AA$ as asymptote at infinite distance. Named by John Bernoulli by analogy with the hyperbola-and-asymptote.
• Lemniscate spiral $z=a\sqrt{n^2-s^2}$ (§527, figure 110). Bounded; figure-eight shape $ACBCA$ tangent to lines $EF$ making angle $n$ with the axis. (Both here.)
• logarithmic-spiral $s=n\log (z/a)$ (§528, figure 111). Equiangular property: every radius from the center meets the curve at the same angle, $\arctan n$. For $n=1$ the angle is $\pi /4$ (semi-rectangular logarithmic spiral). The single transcendental-in-the-equation curve Euler singles out.

Chapter structure

§§	Topic	Page
506–508	Definition of transcendental curve	transcendental-curves
509–510	Intercendental: $y=x^{\sqrt{2}}$	intercendental-curves
511	Complex-exponent: $y=\cos \log x$	transcendental-curves
512–514	Logarithmic curve, fig 101	logarithmic-curve
515, 517	Discrete points below asymptote; $y=(-1{)}^x$	discrete-points-below-asymptote
516	Infinite logarithms of every number	infinite-logarithms-paradox
518	$y=x^x$, fig 102	exponential-curves
519	$z^y=y^z$, fig 103	implicit-exponential-curve
520	Sine line $y/a=\arcsin (x/c)$, fig 104	sine-line
521	Tangent / secant lines	tangent-and-secant-lines
521–522	Cycloid, fig 105	cycloid
523–524	Epicycloid / hypocycloid, fig 106	epicycloid-and-hypocycloid
525	$y=\arccos \log z$, fig 107	arccos-log-curve
526	Spirals general; Archimedes, fig 108–109	spirals, spiral-of-archimedes
527	Hyperbolic and lemniscate, fig 110	hyperbolic-and-lemniscate-spirals
528	Logarithmic spiral, fig 111	logarithmic-spiral

Why the chapter sits at the end

The whole programme of Book II was the systematic study of algebraic curves: orders, asymptotes, species, configurations, tangents, curvature, symmetry, intersection, construction. Transcendental curves require analysis of the infinite (the subject of Book I) for their proper treatment — limits, series, integration — so Euler treats them only as a guided tour, deferring the analytic machinery. The chapter is descriptive and example-driven rather than systematic. The book closes (chapter 22 + appendices) with surfaces, solid geometry, and miscellaneous topics.

Figures

Figures 99–102

Figures 103–105

Figures 106–108

Figures 109–110

Figures 111–114

Related pages

• algebraic-and-transcendental-curves — chapter 1’s brief introduction, here developed in full.
• continuous-and-discontinuous-curves — chapter 1’s law of continuity, here strained by the discrete-points paradox.
• curves-from-polar-coordinates — chapter 17’s polar setup, here reused for the spirals.