Construction of Equations

Summary: The master idea of chapter 20 — inverse of chapter 19. Given a polynomial equation in $z$, find two algebraic curves $F(z,y)=0$ and $G(z,y)=0$ whose simultaneous intersection points, projected onto the axis, have abscissas equal to the real roots of the equation. The construction is most useful when the equation’s roots are needed as line segments; once two suitable curves are drawn, the abscissas of their crossings are read off geometrically.

Sources: chapter20 (§§486–487, §505).

Last updated: 2026-05-12.

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The program (§§486–487)

Chapter 19 set up the forward problem: given two curves, eliminate $y$ to get a polynomial in $z$ whose real roots are the abscissas of their real intersection points (intersection-of-two-curves). Chapter 20 inverts the direction:

Given an algebraic equation $\Phi (z)=0$, find two curves $F(z,y)=0$ and $G(z,y)=0$ such that eliminating $y$ between them yields $\Phi (z)=0$ (possibly times an irrelevant factor).

When such a pair $(F,G)$ is found, the two curves are drawn against the same axis and origin, their intersection points $M,m,m^{\prime },\dots$ are noted, and perpendiculars dropped from each intersection to the axis give abscissas $AP,Ap,A{p}^{\prime },\dots$ which are exactly the real roots of $\Phi$.

The method is “most useful when the roots of some equation are to be expressed by lines” (§486) — i.e., a geometric realization of the roots, in the tradition of Cartesian construction problems.

Two caveats (§§486, 505)

1. Spurious roots. If $\mathrm{deg}\Phi <mn$ (the joint degree of $F,G$), the eliminant carries extra factors; some intersection abscissas will not be roots of $\Phi$. They are easy to recognize and discard once $\Phi$ is known.

2. Missing roots. Conversely a real root $z_0$ of $\Phi$ may correspond to a complex intersection of $F$ and $G$ — an eliminant root whose recovered $y$ is complex — and so produce no visible intersection on the page. The standard remedy is to choose at least one of the two curves of the form $P(z)+Q(z)y=0$ so that $y=-P/Q$ is a single-valued rational function of $z$; then no complex ordinates arise and every real root of $\Phi$ becomes a real intersection. This is the rationale behind the parabola-and-conic construction.

Reading the chapter at a glance

The chapter develops four sample constructions and one general method:

Equation in $z$	Two curves	§§	Figure
linear $Az+B=0$	two lines	§488	97
quadratic $A{z}^2+Bz+C=0$	line + circle (preferred)	§§489–491	98
quadratic	two circles (alternative)	§492	99
biquadratic $z^4+B{z}^2-Cz+D=0$	circle + parabola	§§493–495	100
cubic $z^3+Bz-C=0$	line + parabola ($E=0$ limit)	§495	—
any degree	parabola + conic, via $P+Qy=0$	§§499–504	—

The systematic degree bound $mn$ governs what is possible: a curve of order $m$ and a curve of order $n$ intersect in at most $mn$ points, so an equation of degree $N$ can always be matched by choosing $mn\ge N$ — but the best matches choose $mn=N$ (or factorizations of a slightly larger integer if $N$ is awkward).

Why this matters

The chapter is the bridge between Euler’s Introductio and the broader theory of algebraic curves. It frames the problem solved by Bézout in projective generality: two curves of orders $m$ and $n$ intersect in exactly $mn$ points (counted with multiplicity, including complex and at infinity). Euler’s working out of complex intersections in chapter 19 plus the construction technique here together cover the affine real side of the theorem.

Historically, the chapter also ties back to a classical Cartesian agenda: solve high-degree equations geometrically by sliding rulers and templates against each other. Euler’s contribution is to organize this as a systematic algebraic procedure rather than a case-by-case art.

Related pages

• chapter-20-on-the-construction-of-equations
• intersection-of-two-curves
• complex-intersections
• quadratic-by-line-and-circle
• quadratic-by-two-circles
• biquadratic-by-circle-and-parabola
• intersection-product-degree-bound
• general-construction-by-parabola-and-conic