Higher-Order Surface Cones

Summary: §§129–130 of the Appendix on Surfaces. Brief preview of how the asymptotic-cone machinery extends to third-order and higher surfaces. The third-degree terms either factor (yielding asymptotic planes plus possibly an embedded second-order cone) or not (yielding a genuine third-order cone, of which there are many distinct species — far more variety than the single right/scalene split of second-order cones). Closing observation: no third-order surface lies in a bounded region, just as no third-order curve does.

Sources: appendix5, §§129–130.

Last updated: 2026-05-12.

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§129 — Third-degree homogeneous part

For a third-order surface, the analogous starting point is the homogeneous degree-3 part:

$$\alpha z^3+\beta y{z}^2+\gamma y^{2}z+\delta x{z}^2+ϵxyz+\zeta x^{2}z+\cdots$$

First check: can this be written as a product of linear factors?

• No: the third-degree expression itself is the third-order asymptotic cone. Setting it to zero gives a 3D cone (with vertex at origin) which the surface approaches at infinity.
• Yes: factor into linear pieces — each real factor gives an asymptotic plane.

There are several different third order cones, and from this diversity we obtain several different genera. Although all second order cones are assigned to the same genus, there are right and scalene cones. Still, in the third order there is a much greater variety of cones. (source: appendix5, §129)

§130 — Cases for the third-order homogeneous part

The third-degree expression has degree 3 as a polynomial. Over the reals, the polynomial in three variables has the following possibilities (modulo coordinate choice):

• One real linear factor: the leftover quadratic factor is either a real second-order cone (giving two asymptotes — a plane + a cone) or a complex one (only the plane asymptote).
• Three real linear factors: gives three asymptotic planes. Distinguishing whether all three are distinct, two coincident, or all coincident yields multiple sub-genera.
• One real + two complex linear factors: only one real asymptotic plane.

If all three linear factors are real, depending on whether two or all three are equal to each other, we have two more genera. There are no third order surfaces which lie in a bounded region. (source: appendix5, §130)

No bounded third-order surface

The closing observation: every plane section of an odd-order surface is an odd-order plane curve, and every odd-order plane curve is unbounded (it has at least one branch going to infinity, by branches-at-infinity). Hence no surface of odd order can be everywhere bounded. The third-order case has no analogue of the ellipsoid.

For even orders ≥ 4, bounded surfaces do exist (the quartic Cartesian ovals, for example, have bounded versions). But Euler stops here — he does not pursue the higher-order classification.

Cross-references

• The 3D lift of the analogous remark for curves: no odd-order curve is bounded (branches-at-infinity, chapter-3-on-the-classification-of-algebraic-curves-by-orders).
• The third-order cone variety is the surface analogue of cubic-species-classification — both inherit much more diversity than the second-order case allows.
• The §130 preview is the only mention of higher-order surfaces in Book II — a curtain raised on what would later become Theoria motus corporum solidorum and the calculus-of-surfaces literature of the next century.

Related pages

• appendix-5-on-second-order-surfaces
• asymptotic-cone
• branches-at-infinity
• cubic-species-classification
• chapter-3-on-the-classification-of-algebraic-curves-by-orders
• appendix-6-on-the-intersection-of-two-surfaces