Planar Intersection Condition

Summary: §§151–152 of the Appendix on Surfaces. When does the intersection of two surfaces lie in a plane? Iff both equations satisfy a common linear relation $\alpha z+\beta y+\gamma x=f$. The practical test: solve each surface equation for one variable in terms of the others (say $z=P,y=Q$) and look for a constant $n$ such that $P+nQ$ has only first-degree-in-$z$ and constant terms. Worked example (§152): right cone $z^2=x^2+y^2$ and elliptic hyperboloid $z^2=x^2+2y^2-2ax-a^2$ → subtract to get $y^2=2ax+a^2$ and $z=x+a$, so the intersection lies in the plane $z=x+a$. Closes Book II with the remark that further such investigations await calculus.

Sources: appendix6, §§151–152.

Last updated: 2026-05-12.

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§151 — The general condition

The intersection of two surfaces is generically a non-planar space curve. But it can happen that the intersection accidentally lies in a plane. This is exactly the case when both surface equations together imply a single linear equation $\alpha z+\beta y+\gamma x=f$.

This will be the case if both equations for the surfaces, taken together, satisfy an equation $\alpha z+\beta y+\gamma x=f$. (source: appendix6, §151)

Practical test: solve each surface equation for two of the variables in terms of the third — say $z=P(x),y=Q(x)$ from the two equations. Look for a constant $n$ such that

$$P+nQ=mx+k$$

(linear in $x$, with no $x^2$ or higher powers). If such an $n$ exists, then $z+ny=mx+k$ is the desired linear relation, and the intersection lies in this plane.

§152 — Worked example

Right cone $z^2=x^2+y^2$ and elliptic hyperboloid $z^2=x^2+2y^2-2ax-a^2$. Setting the two right-hand sides equal:

$$x^2+y^2=x^2+2y^2-2ax-a^2,$$

which simplifies to

$$y^2=2ax+a^2.$$

This is a parabola in the $xy$-plane. Substituting back into the cone equation:

$$z^2=x^2+(2ax+a^2)=(x+a{)}^2,$$

so $z=\pm (x+a)$. The ”+” branch gives $z=x+a$.

This last equation already shows that the whole intersection lies in a plane, whose position is given by the equation $z=x+a$. (source: appendix6, §152)

So the intersection is the parabola $y^2=2ax+a^2$ lying in the (tilted) plane $z=x+a$ — a planar curve in space.

Closing remark of Book II

With this kind of reasoning many questions concerning the nature of surfaces can be answered. However, those questions which require more than the method given here, must wait for analysis of the infinite. These two books have prepared the way for that science. (source: appendix6, §152)

This is the final paragraph of the Appendix on Surfaces and thus of Book II. “Analysis of the infinite” is the calculus proper — what Euler will treat in his Institutiones calculi differentialis and integralis. With this remark Euler signals that everything in the Introductio has been pre-calculus — algebra, geometry, and infinite series in service of preparing the reader for differentiation and integration.

Why the example works: a degenerate intersection

The cone and hyperboloid happen to share a common plane through their intersection. Modern observation: subtracting their equations cancels the $z^2$ and $x^2$ terms, leaving a linear-in-$x$ relation in $y$. Geometrically, the elliptic hyperboloid is the cone’s level set offset by a quadratic translation, and the level set happens to be a parabolic cylinder $y^2=2ax+a^2$. The two surfaces meet exactly along this parabolic cylinder’s trace on the plane $z=x+a$.

Cross-references

• The general “linear combination of two equations” trick is reminiscent of the indeterminate-multiplier-elimination method in Book II Chapter 19 §§483–485 — both find linear combinations that simplify a system.
• A planar intersection means the projection onto the appropriate coordinate plane bijects onto the curve, so the $mn$ bound in bezout-bound-surfaces is realized exactly (with $mn=4$ here, but the parabola is degree 2 in its plane — the bound is met because the intersection-curve has algebraic degree 2 in its plane, which lifts to a 4-degree projection in a generic plane).
• The closing reference to “analysis of the infinite” is to the calculus proper. This is the literal final paragraph of Book II.

Related pages

• appendix-6-on-the-intersection-of-two-surfaces
• intersection-of-two-surfaces
• bezout-bound-surfaces
• indeterminate-multiplier-elimination
• non-planar-curves-by-projection
• tangency-of-surfaces