Chapter 8: Concerning Asymptotes

Summary: Chapter 7 catalogued which asymptotic curves arise from each multiplicity profile of real linear factors of the highest member $P$ but deferred their shape. Chapter 8 closes the loop in two complementary directions. (a) For every straight-line asymptote already found, refine it: take the simple-factor case $P=(ay-bx)M$, drop into rotated coordinates $(t,u)$, substitute $u=c$ everywhere except the leading term, and read off a sequence of curvilinear asymptotes $(u-c)+A/t^k=0$ that approach the curve faster than the straight line itself. (b) Repeat the operation for double, triple, and higher factors, getting in each case a complete sub-classification of the resulting branches by $z=C/t^k$ shapes. The chapter ends with a worked 8-branch curve $y^3{x}^2(y-x)-xy(y^2+x^2)+1=0$ that exercises every case (single, double, triple factor) and a final §218 dichotomy: every infinite branch of an algebraic curve is either hyperbolic (converges to a straight line) or parabolic (does not), with infinitely many species in each class.

Sources: chapter8, figures35-39 (figures 35, 36, 37, 38, 39), figures40-43 (figures 40, 41, 42, 43).

Last updated: 2026-04-28

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Why this chapter

chapter-7-on-the-investigation-of-branches-which-go-to-infinity showed that a real linear factor $ay-bx$ of the highest member $P$ of $F=P+Q+R+\cdots =0$ produces an asymptote whose order matches the multiplicity of the factor. Multiplicity 1 gives a straight line (rectilinear-asymptote-from-equation); multiplicity 2 gives a parabola (parabolic-asymptote); multiplicity $\ge 3$ gives a higher-order asymptotic curve (curvilinear-asymptote). Two questions remain:

1. Refinement. Given a straight-line asymptote $u=c$, can we find a curve that approximates the original even more closely? Euler observes (§198) that whenever a straight line is an asymptote, there is always a curve with the same straight line as its own asymptote that hugs the original curve more tightly — capturing not just the limiting direction but also the side of approach (above/below) and any oscillation about the line.
2. Shape catalogue. The asymptotic curves of curvilinear-asymptote are abstract algebraic equations in $(t,u)$. What do they actually look like as $t\to \infty$? In which quadrants do their branches lie? Do they cross the asymptote, lie always to one side, or oscillate?

Chapter 8 answers both. The refinement is a single algebraic move (substitute $u=c$ into all but the leading term, repeat); the shape catalogue is a quadrant analysis of the canonical forms $z=C/t^k$ and $z^2=C/t^k$.

Structure of the chapter

§§198–199 — programme. Asymptotes come from individual factors of $P$; classify by multiplicity; degree of the equation is $n$.

§§200–203 — single linear factor: refinement to curvilinear asymptotes. curvilinear-asymptote-refinement.

• §200: take $P=(ay-bx)M$, rotate axes by $\arctan (b/a)$ to $t=(ax+by)/g$, $u=(ay-bx)/g$ where $g=\sqrt{a^2+b^2}$. The new ordinate $u$ is itself a factor of $P$ (figure 35).
• §201: substitute $y=(au+bt)/g$, $x=(at-bu)/g$ into $P+Q+R+\cdots =0$; expand each member in descending powers of $t$ with coefficients in $u$.
• §202: at infinity the leading equation $\alpha t^{n-1}u+\beta t^{n-1}=0$ gives the straight-line asymptote $u=c$ where $c=-\beta /\alpha$. Substituting $u=c$ everywhere except the first term yields the refinement sequence $(u-c)+A/t=0,(u-c)+A/t^2=0,\dots ,(u-c)+A/t^{n-1}=0$.
• §203: the canonical curvilinear asymptote shape $z=C/t^k$ with $z=u-c$. Quadrant analysis: $k$ odd → branches in opposite quadrants $P,S$ (figure 36); $k$ even → branches in adjacent quadrants $P,Q$ (figure 37).

§§204–209 — double linear factor: detailed refinement. double-factor-asymptote-cases.

• §204: $P=(ay-bx{)}^{2}M$ gives, depending on the leading terms of $Q,R$, either
  • I. $\alpha u^2+\beta t=0$ (parabola, figure 38); or
  • II. $\alpha u^2+\beta u+\gamma =0$ (parallel lines, complex, or double).
• §205: equation II with two real distinct roots $u=c,d$ → two parallel-line asymptotes; refinement of each by the §202 procedure.
• §206: equation II with double root $u=c$ → refinement gives $(u-c{)}^2+A/t^k=0$ for $k=1,2,\dots ,n-2$.
• §207: subtler cases when $A$ vanishes — equation $(u-c{)}^2+A(u-c)/t^p+B/t^q=0$ where $p<q$, $q\le n-2$.
• §208: general form $z^2-Az/t^p+B/t^q=0$ with three sub-cases: $q>2p$ (two distinct curvilinear asymptotes, four branches), $q=2p$ (real ⇔ $A^2>4B$; gives two similar asymptotes $z=C/t^p$), $q<2p$ (middle term dominates near infinity, $z^2+B/t^q=0$).
• §209: shape catalogue for $z^2=C/t^k$. With straight asymptote $XY$ as axis: $k$ odd → branches in $P,R$ (figure 39); $k$ even and $C>0$ → four branches in all four quadrants $P,Q,R,S$ (figure 40); $k$ even and $C<0$ → no real branch.

§§210–214 — triple linear factor: detailed refinement. triple-factor-asymptote-cases.

• §210: $P=(ay-bx{)}^{3}M$. The four leading-equation types depending on which coefficients of $Q,R,S$ are present:
  • I. $\alpha u^3+\beta t^2=0$.
  • II. $\alpha u^3+\beta tu+\gamma t=0$.
  • III. $\alpha u^3+\beta u^2+\gamma t=0$.
  • IV. $\alpha u^3+\beta u^2+\gamma u+\delta =0$.
• §211: I is a third-order line, two branches in $P,Q$ quadrants (figure 41). II resolves into a parabola plus a (refinable) straight line.
• §212: III becomes the cubic $\alpha u^3+\gamma t=0$ (figure 42, two branches in $P,S$). IV gives up to three parallel straight lines, each refinable.
• §213: equal-root sub-cases of IV — double root reduces to the §207 family; triple root produces $(u-c{)}^3+a/t=0$ or the general $(u-c{)}^3+A(u-c{)}^2/t^p+B(u-c)/t^q+C/t^r=0$.
• §214: shape catalogue. The cubic $(u-c{)}^3=K/t^k$ — $k$ odd → opposite quadrants (figure 36 again); $k$ even → adjacent quadrants (figure 37 again).

§215 — beyond multiplicity 3 the same logic continues, and Euler stops. Final example.

§§215–217 — worked example. example-curve-eight-branches: $y^3{x}^2(y-x)-xy(y^2+x^2)+1=0$ with three multiplicities (single $y-x$, double $x^2$, triple $y^3$) and 8 branches at infinity, drawn as figure 43.

§218 — final classification. hyperbolic-and-parabolic-branches. Hyperbolic branches converge to a straight line; parabolic branches do not. Each class has infinitely many species, indexed by exponents in $u^j=A/t^k$ (hyperbolic) and $u^j=A{t}^k$ (parabolic).

Cross-cutting techniques

Substitute-in-all-but-the-leading-term

§202, §205, §206, §211, §213. The single algebraic move that does all the work in this chapter. To refine an asymptote $u=c$, replace $u$ by $c$ everywhere except the term that vanishes when $u=c$ (the term containing $u-c$). What remains is an equation of the form $(u-c)\cdot \alpha t^{n-1}=-F(t)$, where $F(t)$ is a polynomial in $t$ and the constants $\beta ,\gamma ,\delta ,\dots$. Truncating at the first non-vanishing power of $t$ yields the refined equation. Repeated application produces the entire sequence of refinements.

Quadrant analysis of $z=C/t^k$

§203, §209, §214. With the straight-line asymptote as axis, the curve in rotated $(t,z)$ coordinates is read off by sign analysis:

• $z=C/t^k$ — odd $k$ flips sign with $t$, so branches are in diagonally opposite quadrants (figures 36, 41 for cubic, 39 for $z^2=C/t$). Even $k$ preserves sign of $z$, so both branches are on the same side of the asymptote (figure 37 for $z=C/t^2$).
• $z^2=C/t^k$ — same parity rule but doubled. For $k$ even and $C>0$, $z=\pm \sqrt{C}/t^{k/2}$ on both sides of the asymptote in all four quadrants (figure 40).

Polynomial division by $u-c$

§§207, 213. When the next-order coefficient itself contains $u-c$ as a factor, polynomial division of the leading equation by $(u-c{)}^k$ shifts attention to the next coefficient and produces a mixed equation with both $(u-c{)}^k$ and $(u-c{)}^{k-1}$ terms. This is how the equations $(u-c{)}^2+A(u-c)/t+B/t^2=0$ and analogous higher-multiplicity forms arise.

Notable points

• The straight-line asymptote is a degeneracy. §198 reframes the chapter-7 result: every straight-line asymptote is the limiting case of a more informative curvilinear asymptote. The straight line tells you the limiting direction; the curvilinear refinement tells you the side, the rate, and any crossings.
• Refinement always terminates. §202 ends with the alarming-sounding observation: if every refinement coefficient vanishes, then $u-c=0$ would be a part of the curve (the original equation would be divisible by $u-c$), making the curve complex in the sense of complex-curves. So for an irreducible curve, the refinement chain $(u-c)+A/t^k=0$ always stops at some finite $k\le n-1$ with $A\ne 0$.
• Three sub-cases by parity. Whenever the asymptotic equation factors as a square $z^2=C/t^k$, the sign analysis splits into three: $k$ odd gives two branches; $k$ even with $C>0$ gives four; $k$ even with $C<0$ gives none. This trichotomy plays the role for refined asymptotes that the conic trichotomy played for $y^2=\alpha +\beta x+\gamma x^2$ in classification-of-conics.
• Branches need not pair. Where a hyperbola has 2 + 2 = 4 branches and a parabola has 2 branches, a higher-order curve can have any even number — and the example in §§215–217 has 8. The number of branches at infinity is the sum over factors of $P$ of the species-specific count: 2 per simple straight-line refinement, 2 or 4 per double-factor case, 2 per cubic asymptote, etc.
• The §218 dichotomy is exhaustive. Every asymptotic-curve form arrived at in chapters 7 and 8 has the shape $u^j=A/t^k$ or $u^j=A{t}^k$ (after centering $u$ on the appropriate constant). The first family is hyperbolic — the curve approaches the line $u=0$ as $t\to \infty$ — and the second is parabolic — the curve recedes from $u=0$ as $t\to \infty$. There is no third class.
• Foreshadowing infinitesimal calculus. The refinement procedure is precisely a Taylor expansion of $u$ in powers of $1/t$ about $t=\infty$. Euler does it without the language of differentials, but the algebraic content is identical: $u(t)=c+A/t+B/t^2+\cdots$, and chapter 8 is about reading the first non-vanishing correction to the constant value $c$.

What this buys for the rest of Book II

• Chapter 9 onwards classifies higher-order curves species by species — the cubic species in particular (already initiated by Newton), where each cubic is identified by its asymptotic profile drawn from chapters 7 and 8.
• Chapter 8 closes the asymptotic theory of algebraic curves for Book II. Subsequent chapters use the catalogue without extending it further.
• For modern readers, this chapter is the analytic precursor of the theory of Newton polygons and Puiseux expansions: every infinite branch admits a fractional-power expansion in $1/t$, and the leading exponent classifies the branch type.

Figures

Figures 35–39

Figures 40–43

Related pages

• chapter-7-on-the-investigation-of-branches-which-go-to-infinity
• branches-at-infinity
• rectilinear-asymptote-from-equation
• parabolic-asymptote
• curvilinear-asymptote
• curvilinear-asymptote-refinement
• double-factor-asymptote-cases
• triple-factor-asymptote-cases
• example-curve-eight-branches
• hyperbolic-and-parabolic-branches
• asymptotes-of-hyperbola
• complex-curves
• order-of-an-algebraic-curve