Hyperbolic and Parabolic Branches
Summary: The §218 dichotomy that closes chapter-8-concerning-asymptotes and the entire asymptotic theory of Book II. After the multiplicity-by-multiplicity catalogue of chapters 7 and 8, every infinite branch of an algebraic curve falls into exactly one of two classes. Hyperbolic branches converge to a straight-line asymptote — typified by the hyperbola — and have asymptotic equations for positive integers . Parabolic branches do not converge to any straight line — typified by the parabola — and have asymptotic equations . Each class contains infinitely many species, indexed by the integer pair . Each species generally produces two branches at infinity, but with a parity caveat: when both and are even, the equation may have no real solution (no branches) or two distinct real solutions (four branches).
Sources: chapter8 (§218).
Last updated: 2026-04-28
The dichotomy
§218. Every branch of an algebraic curve that goes to infinity is either:
- Hyperbolic: converges to a straight-line asymptote — the curve gets arbitrarily close to a fixed line as the branch recedes.
- Parabolic: does not converge to any straight line — the curve recedes from every line.
The names come from the two paradigmatic conics: the hyperbola has hyperbolic branches (each branch hugs an asymptote line), while the parabola has parabolic branches (the parabola never approaches a straight line; it merely recedes).
Both classes contain infinitely many species, and chapters 7–8 have given the algebraic recipe for each. The species are indexed by the canonical asymptote-equation form, taken in rotated coordinates where is the candidate straight-line asymptote.
Hyperbolic species
A hyperbolic species is given by an equation of the form
with a non-zero constant. The branch lies on the curve , going to as . The line is the straight-line asymptote.
Euler tabulates the species in §218 as a doubly-infinite array:
u = A/t, & u = A/t^2, & u = A/t^3, & u = A/t^4, & \ldots \\ u^2 = A/t, & u^2 = A/t^2, & u^2 = A/t^3, & u^2 = A/t^4, & \ldots \\ u^3 = A/t, & u^3 = A/t^2, & u^3 = A/t^3, & u^3 = A/t^4, & \ldots \\ \vdots \end{array}$$ The first row, $u = A/t^k$, comprises the *simple-factor* refinements of [[curvilinear-asymptote-refinement]]; the second row, $u^2 = A/t^k$, the *double-factor* refinements of [[double-factor-asymptote-cases]]; the third row, $u^3 = A/t^k$, the *triple-factor* refinements of [[triple-factor-asymptote-cases]] §214; and so on. The asymptote is always the line $u = 0$, but the *rate* of approach $u \to 0$ differs by species. ## Parabolic species A parabolic species is given by an equation of the form $$u^j = A t^k, \qquad j > k \geq 1,$$ with $A$ a non-zero constant and $j > k$ to ensure $u \to \infty$ as $t \to \infty$ (so the branch genuinely recedes from every line in the plane). Euler tabulates these as another array: $$\begin{array}{llll} u^2 = At, & u^3 = At, & u^4 = At, & u^5 = At, \ldots \\ u^3 = At^2, & u^4 = At^2, & u^5 = At^2, & u^6 = At^2, \ldots \\ u^4 = At^3, & u^5 = At^3, & u^6 = At^3, & u^7 = At^3, \ldots \\ \vdots \end{array}$$ The first entry $u^2 = At$ is the classical parabola of [[parabola]] (Equation I in [[double-factor-asymptote-cases]]). The first entry of the second row, $u^3 = At^2$, is the cubic asymptote of [[triple-factor-asymptote-cases]] equation I. And so on. ## Branch-count caveat: parity of exponents Each species gives **at least two branches** at infinity, "*provided the exponents of $t$ and $u$ are not both even*". When both $j$ and $k$ are even, two further sub-cases appear: - **No branches**: if the constant $A$ has the wrong sign so that $u^j = A t^k$ admits no real solutions. (For $j = 2, k = 2$: $u^2 = A t^2$ with $A < 0$ has no real root.) - **Four branches**: if the equation has *two distinct real roots* in $u$ for each $t$, doubled by the two signs of $u$. This recovers the case-2 sub-case of [[double-factor-asymptote-cases]] §208 ($q = 2p$, $A^2 > 4B$). For all other parities, two branches per species — one for each sign of the leading coefficient. ## Why the catalogue is exhaustive Every leading-equation form arrived at in chapters 7 and 8 has, after centering on the straight-line asymptote (substituting $z = u - c$), one of these two shapes: - $z^j = A/t^k$ — hyperbolic species — when the curvilinear asymptote *converges* to the line $z = 0$ at infinity. - $z^j = A t^k$ — parabolic species — when the asymptote curve *recedes* from $z = 0$ at infinity. There is no third option: as $t \to \infty$, $z$ either tends to a finite value (hyperbolic) or to infinity (parabolic). The dichotomy is exhaustive because *those are the only two possibilities for an algebraic branch at infinity*. ## Reading the table backwards Given an asymptotic species $u^j = A t^k$ (parabolic) or $u^j = A/t^k$ (hyperbolic), one can ask: from which curve-equation multiplicity profile does this species come? - Hyperbolic $u^j = A/t^k$ comes from a $j$-fold real linear factor of $P$, with the *first non-vanishing* refinement at the $k$-th power of $1/t$. - Parabolic $u^j = A t^k$ comes from the leading equation of a $j$-fold factor with $Q, R, \ldots$ contributing in such a way that $t^k$ appears alone (as in equations I and III of [[triple-factor-asymptote-cases]]). The chapters 7–8 procedure is, equivalently, an *algorithm* that takes a curve $F(x, y) = 0$ and outputs its list of species — one entry per branch at infinity. ## Newton-polygon shadow Modern readers will recognise this as the Newton-polygon analysis of an algebraic curve at the point at infinity, with each (j, k) pair corresponding to a slope on the polygon. Euler's two infinite arrays — hyperbolic and parabolic — are the two halves of the Newton polygon at infinity: hyperbolic species correspond to negative slopes (the curve approaches the axis), parabolic to positive slopes (the curve recedes). The Puiseux expansion of a branch at infinity has leading term $u = A^{1/j} t^{\pm k/j}$, with the sign distinguishing hyperbolic from parabolic. Euler does not phrase any of this in modern algebraic-geometry language, but the algorithmic content is identical, and his §218 table is a complete and correct enumeration of the asymptotic species of an algebraic curve. ## Closure of the asymptotic theory §218 closes the asymptotic study of algebraic curves in Book II. The remaining chapters (9–22 and the appendices) take the catalogue as given and apply it to specific curve families — cubics, quartics, transcendental curves, the algebraic theory of surfaces — without extending the asymptotic theory itself. ## Related pages - [[chapter-8-concerning-asymptotes]] - [[chapter-7-on-the-investigation-of-branches-which-go-to-infinity]] - [[branches-at-infinity]] - [[curvilinear-asymptote-refinement]] - [[double-factor-asymptote-cases]] - [[triple-factor-asymptote-cases]] - [[parabola]] - [[hyperbola]] - [[asymptotes-of-hyperbola]] - [[curvilinear-asymptote]] - [[parabolic-asymptote]] - [[rectilinear-asymptote-from-equation]]