Transcendental Curves

Summary: §§506–508, 511. A curve is transcendental when its defining relation between $x$ and $y$ cannot be brought to polynomial form. Equivalently: $y$ is a transcendental function of $x$, or some exponent in the defining equation is neither integer nor a fraction — a non-irrational form is impossible. Complex exponents may produce real values.

Sources: chapter21 §§506–508, 511.

Last updated: 2026-05-12

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Definition (§506)

A curve which is not algebraic is called transcendental. Hence a transcendental line, which is what such a curve is called, is defined to be one such that the relationship between the abscissa and the ordinate cannot be expressed by an algebraic equation. (source: chapter21, §506)

Equivalently, $y$ is a transcendental function of $x$. Examples already familiar from Book I:

$y=\log x,y=\arcsin x,y=\arccos x,y=\arctan x.$

Any equation in $x,y$ involving these expressions defines a transcendental curve.

How many transcendental curves are there? (§507)

Innumerably more than algebraic ones. Logarithms and circular arcs are only a few examples; analysis of the infinite (the subject of Book I) produces “innumerable other transcendental expressions”. This explains why Euler defers their proper treatment — the systematic tools needed are those of Book I, not Book II.

The “non-irrational” criterion (§508)

An algebraic equation either is non-irrational, with only integral exponents, or is irrational, with fractional exponents. In the latter case the equation can always be expressed in non-irrational form. Any curve whose equation expressing the relation between the coordinates $x$ and $y$ is not non-irrational, or cannot be put into such a form, is always transcendental. (source: chapter21, §508)

In modern terms: a curve is algebraic iff it is the zero locus of some polynomial $F(x,y)\in \mathbb{R}[x,y]$. Even irrational-looking equations like $y=\sqrt{x}$ rationalize to $y^2=x$ — non-irrational. But if an exponent is irrational (e.g. $y=x^{\sqrt{2}}$), no such rationalization is possible: the curve is transcendental. This produces the simplest sub-genus, the intercendental-curves.

Complex exponents (§511)

Imaginary exponents can still produce real-valued curves. The equation

$2y=x^i+x^{-i}$

involves two complex quantities, but their sum is real. Set $\log x=v$, so $x=e^v$; then

$2y=e^{vi}+e^{-vi}=2\cos v,$

using the identity $\frac{e^{vi}+e^{-vi}}{2}=\cos v$ from Book I §138. Hence

$y=\cos \log x.$

Worked numerical check: $x=2$ gives $2y=2^i+2^{-i}$, so $y=\cos \log 2=\cos (0.6931471805599)=0.76923890135408$. The arc with length $\log 2$ in a unit circle measures $39°42^{\prime }51^{\prime \prime }52^{\prime \prime \prime }{9}^{\prime \prime \prime \prime }$, and its cosine produces the listed $y$.

The same curve, inverted to $y=\arccos \log z$, recurs at §525 as the arccos-log-curve (figure 107). Curves of this kind, involving both logarithms and circular arcs, are “correctly referred to as transcendental”.

Where this leads

The chapter §§512–528 unfolds a guided tour:

1. The logarithmic-curve $y=a{e}^{x/b}$ (§§512–515) and its paradoxes — discrete-points-below-asymptote (§§515, 517) and infinite-logarithms-paradox (§516).
2. exponential-curves $y=x^x$ (§518), $z^y=y^z$ (§519).
3. The sine-line $y=a\arcsin (x/c)$ (§520), tangent-and-secant-lines (§521), cycloid (§§521–522), epicycloid-and-hypocycloid (§§523–524), and the arccos-log-curve (§525) mixing both.
4. spirals, including the spiral-of-archimedes, the hyperbolic-and-lemniscate-spirals, and the logarithmic-spiral (§§526–528).

Figures

Figures 106–108

Related pages

• chapter-21-on-transcendental-curves
• algebraic-and-transcendental-curves — chapter 1’s first sketch of the principal division.
• intercendental-curves — the simplest sub-genus, with irrational exponents.