Line–Curve Intersection Bound

Summary: A curve of order $n$ cannot intersect a straight line in more than $n$ points. The proof is a one-liner: choose the straight line as axis, substitute $y=0$ in any equation for the curve, and read off a polynomial of degree at most $n$ in $x$ — whose at most $n$ roots are the intersection abscissas. The bound is tight only generically: complex roots or a vanishing leading coefficient can lower the actual count. The converse fails — $n$ intersections imply order $\ge n$, not $=n$ (§73).

Sources: chapter4

Last updated: 2026-05-12

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Statement (§§66, 72)

A line of the $n^{\text{th}}$ order cannot intersect a straight line in more than $n$ points. (source: chapter4, §72)

Equivalently: the order of an algebraic curve is an upper bound for the number of times any straight line crosses it.

Proof (§67)

Given any algebraic equation $F(x,y)=0$ for the curve, the intersections of the curve with the axis are the points where $y=0$ and $F(x,0)=0$. Setting $y=0$ in $F(x,y)$ leaves a polynomial in $x$ alone whose degree is at most the total degree $n$ of $F$. An $x$-polynomial of degree $\le n$ has at most $n$ roots, so there are at most $n$ intersections (source: chapter4, §§67–68).

The argument applies to any straight line, not only the coordinate axis, because any straight line can be chosen as the axis (§68): this is exactly what coordinate-transformations and degree-invariance permit. The degree of the equation after the coordinate change is still $n$, so the same bound applies.

Why fewer intersections sometimes occur

The bound is $n$; the actual count can be less for three reasons (source: chapter4, §68):

1. Complex roots. The polynomial in $x$ may have some or all roots complex. Complex roots are not geometric intersections.
2. Vanishing leading coefficient. If the coefficient of $x^n$ in $F(x,0)$ is zero, the polynomial has effective degree less than $n$, and fewer roots.
3. Vanishing constant term, etc. — corresponding to intersection at a vertex or a degenerate configuration.

Euler’s case-by-case for low orders

Order 1 (§69) — the straight line $\alpha +\beta x+\gamma y=0$. Setting $y=0$ gives $\alpha +\beta x=0$: one root, one intersection. If $\beta =0$, the reduced equation is $\alpha =0$, which is false (unless $\alpha =0$ too) — meaning the two lines are parallel and do not meet (source: chapter4, §69).

Order 2 (§70) — the general conic $\alpha +\beta x+\gamma y+\delta x^2+ϵxy+\zeta y^2=0$ with $y=0$ leaves $\alpha +\beta x+\delta x^2=0$. Two real roots, one double root, or two complex roots — or, if $\delta =0$, possibly one root. In every case, no more than two intersections (source: chapter4, §70).

Order 3 (§71) — the general cubic with $y=0$ leaves $\alpha +\beta x+\gamma x^2+\delta x^3=0$ (after renaming the coefficients as Euler does in §71). Up to three intersections:

• 3 intersections if all roots real and $\delta \ne 0$;
• 2 if $\delta =0$ and the remaining quadratic has two real roots;
• 1 if two roots are complex, or if $\delta =0=\gamma$;
• 0 if $\delta =0=\gamma =\beta$ but $\alpha \ne 0$ (source: chapter4, §71).

Order $n$ (§72) — same pattern: a polynomial of degree $n$ has at most $n$ real roots, and each missing root (complex, or absorbed by a vanishing leading coefficient) removes one intersection.

The converse is false (§73)

From the number of intersections alone, the order cannot be inferred.

From the number of intersections which an arbitrary straight line makes with some given curve, we cannot determine to which order the curve belongs. If the number of intersections is equal to $n$, it does not follow that the curve is a line of order $n$, since it might belong to some higher order. It could be that the curve is not even algebraic, but is transcendental. (source: chapter4, §73)

The safe inference is one-sided: a curve with $n$ intersections cannot be of order less than $n$. So the intersection bound is a lower bound on the order of a curve, and a non-tight one at that. A curve meeting a line in 4 points is at least quartic, but could be a quintic, a sextic, or even transcendental.

To pin down order exactly from intersections, one would need to test against a generic line (or equivalently, multiple lines in general position) — but Euler does not phrase it this way, and chapter 4 stops at the one-line analysis.

Corollary: three collinear points cannot lie on a simple conic

§79 (see curve-through-given-points) invokes this bound in reverse. If five points in the plane are prescribed for a conic and three of them lie on a straight line, there is no simple conic through all five — because a conic meets that line in at most 2 of the 3 collinear points. The “solution” collapses to a reducible equation: a product of two straight lines, one of them through the three collinear points.

This is the chapter’s first concrete application of the intersection bound to a classification problem, and it foreshadows the broader principle that reducibility is forced by configurations that would violate the bound for a simple curve.

Relation to chapter 3

The intersection bound is the geometric shadow of chapter 3’s definition: “order = degree of the equation.” Each direction of the equivalence is exploited in Book II:

• Order controls geometry: degree $n$ ⇒ at most $n$ intersections (this page).
• Geometry controls order (partially): at least $k$ intersections on some line ⇒ order $\ge k$ (§73, above).

See order-of-an-algebraic-curve for the algebraic definition and degree-invariance for the theorem that lets Euler freely change axis in the proof.

Generalization to two curves (chapter 19)

The line–curve case is the $(1,n)$ instance of the general elimination of one coordinate between two simultaneous curve equations. Chapter 19 develops this systematically: substitute the line’s parameterization into the curve as here, more generally eliminate $y$ between two arbitrary equations, and count real roots of the resulting polynomial in $z$. Two new features appear that are absent in the line–curve case:

• complex-intersections — when $y$ is irrational in $z$ after elimination, real roots of the eliminant need not correspond to real intersections; the ordinate may be complex. The line–curve case escapes this entirely because substituting a linear equation gives $y$ as a single-valued rational function of $z$.
• The $mn$ bound on real intersections (foreshadowing Bézout) — the eliminant has $z$-degree at most $mn$, with the actual real-intersection count possibly less, paralleling the three reductions listed above.

See intersection-of-two-curves, elimination-of-ordinate, and indeterminate-multiplier-elimination for the algebraic methods.

Related pages

• chapter-4-on-the-special-properties-of-lines-of-any-order
• order-of-an-algebraic-curve
• degree-invariance
• curve-through-given-points
• complex-curves
• straight-line-equation
• general-equation-of-order-n
• chapter-19-on-the-intersection-of-curves
• intersection-of-two-curves