Chapter 19 — On the Intersection of Curves

Summary: Chapter 19 generalizes the line–curve bound from chapter 4 to two arbitrary curves: their intersections are found by eliminating $y$ from the two simultaneous equations and reading the real roots of the resulting polynomial in $z$. The chapter divides into three movements: the procedure itself (§§457–464); the new phenomenon of complex intersections — real roots of the eliminant that fail to give real intersections because the ordinate value is complex (§§465–473); and two systematic methods for performing the elimination — cascaded subtraction with closed-form tables for $\mathrm{deg}y=1,2,3,4$ (§§474–482) and an alternative via indeterminate multipliers that is essentially a resultant construction (§§483–485).

Sources: chapter19 (§§457–485); figures 94, 95, 96 in figures94-98.

Last updated: 2026-05-12.

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The chapter at a glance

Chapter 4 (line-curve-intersection-bound) established that a curve of order $n$ meets a straight line in at most $n$ points. The proof was algebraic: substitute the line’s parameterization into the curve’s equation and count real roots. Chapter 19 generalizes that move to two curves, neither of which need be linear — and discovers in the process that the elimination is more subtle than it looked. Three new ingredients appear:

1. when both curves are multi-valued in $z$, the eliminated equation can have real roots that correspond to no real intersection (the ordinate is complex);
2. for low-degree pairs $(m,n)$ in $y$, the eliminant has a closed combinatorial form, and Euler tabulates it;
3. for higher-degree pairs, a more systematic method using indeterminate multipliers avoids the doubling complexity of the closed-form approach — this is essentially the resultant.

1. The elimination procedure (§§457–464)

The chapter opens with the line–curve case as warm-up (figure 94, §458). With curve $AMm$ in coordinates $(z,y)$ and straight line $BMm$ given by $\alpha z+\beta y=\gamma$, substitute $y=(\gamma -\alpha z)/\beta$ into the curve’s equation to get a polynomial in $z$ alone whose real roots are the abscissas of the intersections, and recover $y$ at each root from the line’s equation. The degree of the substituted polynomial is at most that of the curve — re-establishing the chapter-4 bound (§460).

Two genuine curves (figure 95, §463) work the same way in principle but require careful elimination: take the same $z$ in both equations, eliminate $y$ to obtain a single polynomial in $z$, find its real roots, and recover the ordinates.

See intersection-of-two-curves.

2. Complex intersections (§§465–473)

Here Euler discovers a phenomenon absent in the line–curve case. When $y$ — recovered as a function of $z$ from the elimination — is irrational in $z$, two ordinate values from one curve at a given $z_0$ may be complex but equal to each other. Then $z_0$ satisfies the eliminant but no real intersection exists at $z_0$: this is a complex intersection.

Three illuminating examples drive the section:

• §465 (parabola vs. circle, sharing axis). $y^2-2zy+z^2-2az=0$ and $y^2+z^2-c^2=0$. Subtracting linearizes $y$ as a rational function of $z$, so every real root of the quartic in $z$ gives a real intersection. No complex intersections here.
• §§467–468 (disjoint parabola and circle, figure 96). $y^2=-2az-2ab$ and $y^2=-2cz-z^2$ with $AE=b$ such that no real intersection exists. Eliminating $y$ produces a quadratic with two real $z$-roots whose corresponding $y$ values are complex — pure complex intersections.
• §§471–473 (cubic vs. parabola with spurious factor). $y^3-3a{y}^2+2a^{2}y-6a{z}^2=0$ and $y^2-2az=0$. After elimination, $y=3z$ is linear in $z$ — but the equation factors as $(y-3z)\cdot (\text{linear in }z)=0$; the second factor gives a real $z$-root whose parabola ordinate is complex. Lesson: linearity of the eliminant in $y$ is not enough — a hidden factor can still admit complex intersections.

Sufficient condition for no complex intersections (§§469–470): after elimination, $y$ should be a single-valued (non-irrational) function of $z$, and the elimination should not have involved a polynomial division by a factor that could mask the irrationality.

See complex-intersections.

3. Closed-form eliminants for low degrees (§§474–482)

Euler tabulates the eliminant for each pair $(m,n)$ of degrees of $y$:

Degrees $(m,n)$	Eliminant in $P,Q,R,S,T,p,q,r,s,t$
$(1,1)$	$pQ-Pq=0$
$(1,2)$	$P^{2}r-PQq+Q^{2}p=0$
$(1,3)$	$Q^{3}p-P{Q}^{2}q+P^{2}Qr-P^{3}s=0$
$(1,4)$	$Q^{4}p-P{Q}^{3}q+P^2{Q}^{2}r-P^{3}Qs+P^{4}t=0$
$(2,2)$ pure even	$Pr-Rp=0$
$(2,2)$ general	$P^2{r}^2-2PRpr+R^2{p}^2+Q^{2}pr-PQqr-QRpq+PR{q}^2=0$
$(2,3)$	long expansion, §480
$(3,3)$	long expansion divided by $Sp-Ps$, §481
$(4,4)$	cascade of brevity letters $A,B,C,D\mid a,b,c,d\mid E,F,G\mid H,I,h,i$, §482

The $(4,4)$ case is a tour de force: four passes of subtraction-and-substitution, each reducing the joint degree by one, with brevity substitutions that exploit the symmetries $a=D$, $G=e$, $I=h$ to close the recursion. The final equation comes out as $Hi-Ih=0$, expanded to a polynomial in just eight letters (four upper, four lower) after observing divisibility by $D^2=a^2$.

See elimination-of-ordinate.

4. The indeterminate-multiplier method (§§483–485)

The cascaded-subtraction method doubles the size of the expression at each step. Euler concludes the chapter with an alternative: multiply each equation by a polynomial in $y$ with indeterminate coefficients $A,B,C,\dots$ and $a,b,c,\dots$, equate the products, and match coefficients of every power of $y$. The matching equations form a linear system in the indeterminates; the final (coefficient of $y^0$) equation, after solving, is the eliminant.

The counting: with both products of degree $k$ in $y$, the indeterminate-letter count $2k-m-n$ must equal the matching-equation count $k-1$, forcing $k=m+n-1$. Euler demonstrates on the $(2,3)$ case and verifies that the result coincides with the $(2,3)$ entry of the §480 table.

In modern terms, this construction produces the resultant ${\mathrm{Res}}_y(F,G)$ as a determinant of the Sylvester matrix — a determinantal condition for the two polynomials in $y$ to share a root. Euler does not phrase it as a determinant, but the algebraic content is identical.

See indeterminate-multiplier-elimination.

Foreshadowing Bézout

The chapter establishes that the eliminant of two equations of joint degrees $(m,n)$ in $y$ has $z$-degree at most $mn$, and that real roots of the eliminant overcount real intersections (because of complex intersections) — so real intersections are at most $mn$. The full Bézout statement that the count is exactly $mn$ in projective space, with multiplicity, including complex points, requires the next chapter to develop in earnest. Chapter 19 supplies the algebraic machinery; chapter 20 will apply it.

Connections to earlier chapters

• The line–curve bound of line-curve-intersection-bound (§§66–73) is the $(1,n)$ case of this chapter’s general elimination.
• The discussion of complex-intersections connects to multiple-points-on-curves (§§281–282): two coincident real roots of the eliminant signal a tangency or self-intersection, while a pair of complex-conjugate values gives a conjugate point.
• The $(2,2)$ pure-even sub-case of elimination-of-ordinate presupposes that the axis is a diameter of both curves — the chapter-15 condition.
• The closed-form (1,1) eliminant $pQ-Pq=0$ is the simplest instance of the line–line intersection: two lines meet in one point unless they are parallel, in which case the eliminant degenerates.

Figures

Figures 94–98

Related pages

• intersection-of-two-curves
• complex-intersections
• elimination-of-ordinate
• indeterminate-multiplier-elimination
• line-curve-intersection-bound
• multiple-points-on-curves
• diameter-and-center-from-equation