Intercendental Curves

Summary: §§509–510. Leibniz’s term for the simplest sub-genus of transcendental-curves: those with irrational exponents, such as $y=x^{\sqrt{2}}$. Algebraic approximations exist (orders 3, 7, 17, 41, 99, …) but the curve itself has infinite order. Exact construction requires logarithms.

Sources: chapter21 §§509–510.

Last updated: 2026-05-12

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Setup (§509)

When the exponent in $y=x^p$ is irrational, no algebraic manipulation can produce a polynomial equation: the curve is transcendental. Since these curves involve “neither logarithms nor circular arcs, but simply the irrationality of exponents, they seem to pertain more to geometry”, Leibniz called such functions intercendental — as if situated between the algebraic and the transcendental (§508).

The canonical example:

$y=x^{\sqrt{2}}.$

Why approximation fails (§509)

Approximate $\sqrt{2}$ by convergents from the sequence $3/2,7/5,17/12,41/29,99/70,\dots$ Each gives an algebraic curve, but the order of the curve equals the rationalization degree:

• $y=x^{3/2}\Rightarrow y^2=x^3$ — order $3$.
• $y=x^{7/5}\Rightarrow y^5=x^7$ — order $7$.
• $y=x^{17/12}\Rightarrow y^{12}=x^{17}$ — order $17$.
• $y=x^{41/29}$ — order $41$.

As the approximation improves, the order grows without bound. Since $\sqrt{2}$ has no finite rational expression, the limit curve has infinite order — it cannot be algebraic.

A second subtlety: $\sqrt{2}$ is two-valued ($+\sqrt{2}$ and $-\sqrt{2}$), so $y$ has two values for each positive $x$ — the curve is really a pair of curves.

Exact construction by logarithms (§510)

Direct point-plotting requires logarithms. Take logs:

$\log y=\sqrt{2}\cdot \log x.$

For each $x$, look up $\log x$, multiply by $\sqrt{2}=1.41421356$, and look up the antilog. Sample values:

$x$	$\log x$	$\sqrt{2}\log x$	$y$
0	$-\infty$	$-\infty$	$0$
1	0	0	1
2	0.3010300	0.4257274	2.665186
10	1	1.4142356	25.955870

For negative $x$, even $(-1{)}^{\sqrt{2}}$ has no clear definition — the approximations to $\sqrt{2}$ are no help. The curve is essentially defined only for positive $x$.

Place in the chapter

Intercendental curves are §508’s first species of transcendental: the simplest case, because they involve no analytic transcendence (no $\log$, no $\arcsin$, no series) — just an irrational power. They are also the most geometrically inaccessible, since rational exponents are the only ones that admit a finite construction by ruler-and-compass methods.

Related pages

• transcendental-curves — broader definition and the complex-exponent §511 example.
• chapter-21-on-transcendental-curves
• order-of-an-algebraic-curve — what the “infinite order” claim means.