Chapter 7: On the Investigation of Branches Which Go to Infinity

Summary: Having classified second-order lines into ellipse/parabola/hyperbola in chapter-6-on-the-subdivision-of-second-order-lines-into-genera by reading off behavior at infinity from the leading coefficients, Euler now generalises to algebraic curves of any order. The principle is unchanged but more powerful: writing the equation as $P+Q+R+S+\cdots =0$ in members of descending degree, the highest member $P$ alone decides whether and how the curve escapes to infinity. Branches at infinity exist iff $P$ has a real linear factor of the form $ay-bx$; the multiplicity of such factors determines the type of asymptote — straight line (multiplicity 1), parabolic (multiplicity 2), curvilinear of order $k$ (multiplicity $k$). The chapter is a complete catalog through multiplicity 4, with a general principle stated for higher multiplicities. The actual shape of the resulting curvilinear asymptotes is deferred to chapter 8.

Sources: chapter7

Last updated: 2026-04-28

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Why this chapter

chapter-6-on-the-subdivision-of-second-order-lines-into-genera showed that for conics, the sign of $\gamma$ in $y^2=\alpha +\beta x+\gamma x^2$ — equivalently, the discriminant of the highest member $\alpha y^2+\beta xy+\gamma x^2$ — partitions conics into three genera by the number of branches at infinity. Chapter 7 generalises:

• Replace “second-order line” with “algebraic curve of any order $n$“.
• Replace “discriminant of the highest member” with “factorisation pattern of the highest member into real linear factors and irreducible real quadratic factors”.
• The output is no longer a 3-way classification but a richer hierarchy organised by the multiplicity profile of the real linear factors.

Structure of the chapter

§§166-169 — branches at infinity exist iff $P$ has a real linear factor. branches-at-infinity.

• §166: branch at infinity ↔ at least one of $x,y$ becomes infinite.
• §167: linear-factor decomposition of $P$; complex factors pair into real quadratic factors $A^2{y}^2-2ABxy\cos \varphi +B^2{x}^2$ which are strictly $>0$ off the origin and never vanish at infinity.
• §168-169: contrapositive — no real linear factor in $P$ implies no infinite branch (curve bounded). The ellipse case for $n=2$.

§§170-177 — straight-line asymptotes. rectilinear-asymptote-from-equation.

• §170: members $P,Q,R,S,\dots$ by descending degree.
• §171-172: $P=(ay-bx)M$ with $M$ degree $n-1$ free of real linear factors. At infinity, $ay-bx$ stays bounded; set it equal to a finite constant $p$.
• §173: explicit formula $p=-Q/M$ evaluated at $y/x=b/a$ (substitute $y=b$, $x=a$).
• §174: asymptote $ay-bx=p$ is a straight line; two opposite branches.
• §175-177: two distinct real linear factors → two distinct straight asymptotes, four branches → hyperbola.

§§178-184 — parabolic asymptote (double factor). parabolic-asymptote.

• §178-179: $P=(ay-bx{)}^{2}M$. After axis rotation $t=(ax+by)/g$, $u=(ay-bx)/g$ with $g=\sqrt{a^2+b^2}$, the asymptotic equation is $u^2+At/g+B/g^2=0$ — a parabola.
• §180: degenerate case $A=0$, $B<0$ — two parallel-line asymptotes, four branches.
• §181: $A=0$, $B>0$ — no infinite branch (bounded direction).
• §182-184: $A=B=0$ — recurse on next member; hyperbolic curvilinear asymptote $u^2=-C/(g^{3}t)$, then $u^2=-D/(g^4{t}^2)$, etc.

§§185-197 — curvilinear asymptotes (triple, quadruple, general factor). curvilinear-asymptote.

• §185: three distinct real factors → three lines, six branches; partial coincidence → reduce to subcases.
• §186-187: $P=(ay-bx{)}^{3}M$ → cubic asymptote $u^3+A{t}^2/g=0$.
• §188-189: $Q$ contains the linear factor → mixed asymptotes (line + cubic, or two lines + parabola).
• §190-192: when leading expansion vanishes, examine successive members. General normal form $(u-\alpha {)}^k=K/t^{\mu }$ for $k$ coincident roots and leading non-vanishing power $\mu$.
• §193-196: $P=(ay-bx{)}^{4}M$ → quartic asymptote and partial-coincidence sub-cases (four parallel lines, parabola pairs, etc.).
• §197: general principle — multiplicity-$k$ factor produces an asymptotic curve of order at most $k$. The variety in bounded regions is the topic of chapter 8.

Cross-cutting techniques

Member decomposition

The single most important move in the chapter is to write $F(x,y)=P+Q+R+S+\cdots$ where $P$ is the homogeneous-of-top-degree part, $Q$ the next, etc. (§170). Behavior at infinity is then read off from $P$ alone, with $Q,R,S,\dots$ providing successive corrections. This is essentially a Taylor-like decomposition of $F$ at the line at infinity — modern algebraic geometry recognises it as the projective closure-and-localisation at infinity, but Euler does it with bare-hands algebra.

Rotating to the asymptote direction

Recurring throughout (§§183, 187, 193). Once a real linear factor $ay-bx$ of $P$ is identified, set

$t=\frac{ax+by}{g},u=\frac{ay-bx}{g},g=\sqrt{a^2+b^2}.$

Then $u$ measures perpendicular displacement from the asymptote direction, $t$ runs along it, and $t\to \infty$ along the branch. The asymptotic equation in $(t,u)$ has clean low-order form, ready for direct interpretation as a parabola, cubic, etc.

Substitute $y/x=b/a$ to evaluate ratios

Each asymptotic constant $A,B,C,\dots$ is of the form $\text{(homogeneous numerator)}/\text{(homogeneous denominator)}$ with equal degrees, hence a function of $y/x$ only. Substituting $y=b,x=a$ — the asymptotic direction — yields a numerical constant. This homogeneity-reduction trick is what makes the explicit formulas like $p=-Q/M{|}_{y=b,x=a}$ workable.

The recursion-on-degeneration logic

Whenever the leading asymptotic constant vanishes, drop to the next member and recur. The recursion always terminates because the equation has finitely many members. The shape of the asymptote is therefore controlled by the first non-vanishing asymptotic constant — and which member that comes from determines the order of the curvilinear asymptote.

Notable points

• One algebraic invariant — many shapes. Where chapter 6 had a 3-way trichotomy (sign of one quantity), chapter 7 has an arbitrarily rich hierarchy: the multiplicity profile of the real linear factors of $P$, refined by the divisibility properties of $Q,R,S,\dots$ by those same factors. An order-$n$ curve can have up to $n$ asymptotes, each of any type from straight line through order-$n$ curvilinear.
• Hyperbola is the ”$k=1$ twice” case. §175-177 derives the hyperbolic shape — two distinct straight asymptotes — purely from $P=(ay-bx)(cy-dx)\cdot \text{const}$ for a second-order line. The classical conic classification falls out of the multiplicity profile $\{1,1\}$.
• Parabola is the "$k=2$" case. §§178-184 generalise: any algebraic curve of any order with a double real linear factor in $P$ has parabolic asymptotic behavior in that direction. The chapter-6 parabola $y^2=2cx$ is just $n=2$ of this general phenomenon.
• The asymptote can be a curve. Euler is explicit (§§186-187) that asymptotes need not be straight lines: a triple linear factor produces a cubic asymptote, a quadruple factor a quartic, and so on. The notion of “curve to which the original curve coincides at infinity” is wider than “straight line approached arbitrarily closely”. This is a real conceptual departure from elementary geometry — modern texts often hide it by speaking only of straight asymptotes.
• Iff fails in one direction. The criterion in branches-at-infinity is that $P$ has no real linear factor ⇒ no infinite branch. The converse fails: §181 exhibits $P=(ay-bx{)}^{2}M$ with $A=0,B>0$ — $P$ has a real factor but the curve has no infinite branch. Real linear factors of $P$ are necessary but not sufficient for infinite branches.
• The chapter prepares chapter 8. The asymptotic curves $u^3+A{t}^2/g=0$, $u^2=K/t$, $(u-\alpha {)}^k=K/t^{\mu }$ etc. arrived at here are new objects that have to be drawn and analysed in their own right. Euler explicitly defers this (§187): “in the next chapter we will consider these asymptotic curves.” Chapter 7 names them; chapter 8 studies them.

What this buys for the rest of Book II

• Chapter 8 picks up the asymptotic curves catalogued here (cubic, quartic, hyperbolic, etc.) and studies their shapes individually.
• Chapters 9 and onwards use the multiplicity-of-linear-factors classification as a primary tool for cataloging higher-order curves. The familiar Newtonian classification of cubics, in particular, fits into this framework: each cubic species is recognised by its asymptotic profile.
• The deeper lesson — that the projective closure of an algebraic curve, restricted to the line at infinity, encodes everything about its asymptotic behavior — would not be made explicit until 19th-century projective geometry. Euler’s chapter-7 calculations are the analytic shadow of that geometric fact.

Related pages

• branches-at-infinity
• rectilinear-asymptote-from-equation
• parabolic-asymptote
• curvilinear-asymptote
• chapter-6-on-the-subdivision-of-second-order-lines-into-genera
• classification-of-conics
• hyperbola
• parabola
• asymptotes-of-hyperbola
• order-of-an-algebraic-curve
• general-equation-of-order-n