Algebraic and Transcendental Curves

Summary: The principal division of continuous curves. A curve is algebraic (also called geometric) if $y$ is an algebraic function of $x$, i.e. the curve is cut out by a polynomial equation in $x$ and $y$; it is transcendental if the defining relation is transcendental.

Sources: chapter1

Last updated: 2026-05-12

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Definition (§15)

A curve will be algebraic if the ordinate $y$ is an algebraic function of the abscissa $x$, or, since the nature of the curve is expressed by an algebraic equation in the coordinates, this kind of curve is also accustomed to be called geometric. A transcendental curve is one whose nature is expressed by a transcendental equation in $x$ and $y$, or an equation in which $y$ is equal to a transcendental function [of] $x$. This is the principal division of continuous curves, whereby they are either algebraic or transcendental. (source: chapter1, §15)

Relation to Book I

The division here is the geometric image of Euler’s Book I classification of functions. The function-side classification (algebraic vs. transcendental, algebraic subdivided into rational/irrational, etc.) transfers verbatim: the curve is algebraic iff its defining function is algebraic, and so on.

Consequences carried over from the function classification:

• A curve defined by $y=P(x)$ with $P$ a polynomial is algebraic.
• A curve cut out by $F(x,y)=0$ with $F$ a polynomial in both variables is algebraic (even when $y$ cannot be solved for in closed form — implicit algebraic).
• A curve defined by $y=\sin x$, $y=\log x$, $y=e^x$, etc. is transcendental.

Why “geometric”

Euler records the alternative name geometric for algebraic curves. This terminology goes back to the 17th century: Descartes admitted as “geometric” exactly the curves with polynomial equations, banishing the rest (spirals, quadratrix, etc.) as “mechanical.” By Euler’s time the Cartesian prohibition had been dropped, but the label persists.

Scope beyond the chapter

Book II proper mostly treats algebraic curves — conic sections, general conics, cubic curves, branches and asymptotes of algebraic curves, and so on — with transcendental curves handled more sparingly and later. The division introduced here is therefore an organizing axis for the entire volume.

The full development of transcendental curves is deferred to chapter 21 — see transcendental-curves for the §§506–508 definition and the wider catalogue, and chapter-21-on-transcendental-curves for the chapter overview.

The Appendix on Surfaces lifts this dichotomy one dimension up in algebraic-and-transcendental-surfaces (§§6–7).

Related pages

• chapter-1-on-curves-in-general
• abscissa-and-ordinate
• continuous-and-discontinuous-curves
• multi-valued-curves
• transcendental-curves — chapter 21’s full development.
• chapter-21-on-transcendental-curves
• algebraic-and-transcendental-surfaces — surface analogue in the Appendix.