Algebraic and Transcendental Surfaces

Summary: §§6–7 of the Appendix on Surfaces. The principal division of regular (continuous) surfaces parallels the algebraic-and-transcendental-curves dichotomy of plane curves: a surface is algebraic if its equation in $x,y,z$ involves only polynomial operations, and transcendental otherwise. Examples of the latter: $z=x\log y$, $z=y^x$, $z=y\sin x$.

Sources: appendix1, §§6–7.

Last updated: 2026-05-12.

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Definition (§7)

A surface is called algebraic if its nature can be expressed by an algebraic equation in the coordinates $x,y,z$, that is, when $z$ is equal to an algebraic function in $x$ and $y$. On the other hand, if $z$ is not equal to an algebraic function in $x$ and $y$, or if in the equation between $x,y$, and $z$ there enter transcendental functions such as logarithms or those dependent on circular arcs, then the surface whose nature is expressed in this kind of equation is called transcendental. (source: appendix1, §7)

Three sample transcendental surfaces:

• $z=x\log y$
• $z=y^x$
• $z=y\sin x$

The treatment in the rest of the Appendix focuses entirely on the algebraic case, just as Book II focused mainly on algebraic curves before chapter 21 brought in transcendentals.

Continuous (regular) vs. irregular (§6)

A surface is continuous or regular if a single equation $z=f(x,y)$ describes every point on it; irregular if the surface is patched together from several functions on different parts (e.g. a sphere joined to a cone, or a sphere joined to a plane). The Appendix treats only continuous surfaces — once those are understood, irregular ones reduce to bookkeeping of pieces.

The Book II analogue

This section is a one-paragraph translation of algebraic-and-transcendental-curves from chapter 1 of Book II. The dichotomy is the same; the operating principle (transcendentals deferred until algebraic surfaces are well understood) is the same; the Appendix never reaches a “chapter 21 of surfaces” treatment of transcendental cases.

Cross-references

• Plane-curve analogue: algebraic-and-transcendental-curves (Book II Chapter 1).
• Continuous-vs-irregular distinction parallels continuous-and-discontinuous-curves (also Book II Chapter 1).
• The chapter-1 sketch of transcendental curves is fully developed in chapter-21-on-transcendental-curves; no analogous “chapter on transcendental surfaces” appears in this Appendix.

Related pages

• appendix-1-on-the-surfaces-of-solids
• surface-coordinates-and-equation
• algebraic-and-transcendental-curves
• continuous-and-discontinuous-curves