Cubic Species Classification

Summary: The four-case enumeration that produces all sixteen species of third-order line, organized by the factor structure of the cubic principal member.

Sources: chapter9 (§§223–235).

Last updated: 2026-04-30.

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Working form

After rotating coordinates so that one real linear factor of the cubic principal member becomes $u$, the equation reads

$\alpha t^{2}u+\beta t{u}^2+\gamma u^3+\delta t^2+ϵtu+\zeta u^2+\eta t+\theta u+\iota =0.$

The principal member factors as $u(\alpha t^2+\beta tu+\gamma u^2)$. The remaining quadratic in $t/u$ has discriminant ${\beta }^2-4\alpha \gamma$, which together with possible higher coincidences gives four exhaustive cases.

Case 1 — single real factor (§224)

Condition: ${\beta }^2<4\alpha \gamma$, so $\alpha t^2+\beta tu+\gamma u^2$ has no real factors and only the direction $u=0$ is real at infinity.

Setting $t\to \infty$ leaves $\alpha u+\delta =0$, i.e. the straight-line asymptote $u=c$ with $c=-\delta /\alpha$. Substituting $u=c$ in the lower-order members gives

$\alpha t^2(u-c)+(\beta c^2+ϵc+\eta )t+(\gamma c^3+\zeta c^2+\theta c+\iota )=0.$

Two refinement rates depending on whether $\beta c^2+ϵc+\eta$ vanishes:

• FIRST species — $u-c=A/t$.
• SECOND species — $u-c=A/t^2$.

Case 2 — three distinct real factors (§§225–227)

Condition: ${\beta }^2>4\alpha \gamma$.

The same refinement applies independently to each factor: each gives an asymptote of the form $u=A/t$ or $u=A/t^2$. Combinatorially this would suggest four signatures $(3,0)$, $(2,1)$, $(1,2)$, $(0,3)$. Euler tests the most general representative,

$y(\alpha y-\beta z)(\gamma y-\delta x)+ϵxy+\zeta y^2+\eta x+\theta y+\iota =0,$

and tracks each factor:

• $y$ gives $u=A/t$ unless $\eta =0$.
• $\alpha y-\beta z$ gives $u=A/t$ unless $\frac{\alpha \eta +\beta \theta }{\beta }+\frac{(\alpha ϵ+\beta \zeta )(\gamma ϵ+\delta \zeta )}{(\alpha \delta -\beta \gamma {)}^2}=0$.
• $\gamma y-\delta x$ gives $u=A/t$ unless $\frac{\gamma \eta +\delta \theta }{\delta }+\frac{(\alpha ϵ+\beta \zeta )(\gamma ϵ+\delta \zeta )}{(\alpha \delta -\beta \gamma {)}^2}=0$.

Whenever the relevant expression vanishes, that factor’s asymptote refines to $u=A/t^2$ instead.

§227 — parameter analysis. Setting $\eta =0$ together with one of the other two expressions vanishing forces the third also to vanish. Therefore a curve cannot have exactly one asymptote of $u=A/t$ and exactly two of $u=A/t^2$; that combination is impossible. The signatures that survive are:

• THIRD species — three asymptotes of $u=A/t$.
• FOURTH species — two of $u=A/t$, one of $u=A/t^2$.
• FIFTH species — three of $u=A/t^2$.

(Euler had tentatively introduced a “fifth” with signature $(1,2)$ and a “sixth” with signature $(0,3)$; the impossibility forces him to drop the $(1,2)$ entry and renumber, so what was the sixth becomes the fifth.)

Case 3 — double factor, third distinct (§§228–232)

The leading $\alpha t^{2}u$ term vanishes (otherwise the principal member would have $u$ as a single factor, not a doubled one). The general equation becomes

$\alpha t{u}^2+\beta u^3+\gamma t^2+\delta tu+ϵu^2+\zeta t+\eta u+\theta =0,$

with principal member $u^2(\alpha t-\beta u)\cdot \text{(constant)}+\gamma t^2\cdot \text{(rest)}$ — the doubled factor is $u$, and the third factor is $\alpha t-\beta u$ (when $\gamma$ behaves accordingly).

The third factor $\alpha t-\beta u$ produces an asymptote of the form $u=A/t$ unless

$(\alpha \delta +2\beta \gamma )({\alpha }^2ϵ+\alpha \beta \delta +{\beta }^2\gamma )-{\alpha }^3(\alpha \eta +\beta \zeta )=0,$

in which case it refines to $u=A/t^2$. This binary alternative is then crossed with three sub-cases for the doubled factor, distinguished by what survives at $t=\infty$.

§229 — $\gamma \ne 0$: parabolic asymptote $u^2=At$

Letting $t\to \infty$ leaves $\alpha u^2+\gamma t=0$, a parabolic asymptote of the species $u^2=At$.

• SIXTH species — third-factor $u=A/t$, plus this $u^2=At$.
• SEVENTH species — third-factor $u=A/t^2$, plus this $u^2=At$.

§§230–231 — $\gamma =0$, with $\alpha u^2+\delta u+\zeta =0$ at infinity

The equation becomes

$\alpha t{u}^2-\beta u^3+\delta tu+ϵu^2+\zeta t+\eta u+\theta =0,$

and at $t\to \infty$ the constraint $\alpha u^2+\delta u+\zeta =0$ governs the doubled factor. Three sub-cases by the discriminant ${\delta }^2-4\alpha \zeta$:

§230 — ${\delta }^2<4\alpha \zeta$ (no real roots — no asymptotic branch from the double factor):

• EIGHTH species — single asymptote $u=A/t$ (from the third factor; condition $\delta (\alpha ϵ+\beta \delta )\ne \alpha (\alpha \eta +\beta \zeta )$).
• NINTH species — single asymptote $u=A/t^2$ (from the third factor; the equation of §228 holds).

§231 — ${\delta }^2>4\alpha \zeta$ (two distinct real roots → two parallel straight-line asymptotes $u=A/t$):

• TENTH species — third-factor $u=A/t$ + the two parallel $u=A/t$.
• ELEVENTH species — third-factor $u=A/t^2$ + the two parallel $u=A/t$.

§232 — ${\delta }^2=4\alpha \zeta$ (equal roots: $\alpha u^2+\delta u+\zeta =\alpha (u-c{)}^2$):

The double-factor equation reduces to $\alpha t(u-c{)}^2=\beta c^3-ϵc^2-\eta c-\theta$, a parabolic asymptote of the species $u^2=A/t$ (i.e. $(u-c{)}^2\cdot t=$ const).

• TWELFTH species — third-factor $u=A/t$, plus this $u^2=A/t$.
• THIRTEENTH species — third-factor $u=A/t^2$, plus this $u^2=A/t$.

Case 4 — all three factors equal (§§233–235)

The equation becomes

$\alpha u^3+\beta t^2+\gamma tu+\delta u^2+ϵt+\zeta u+\eta =0.$

§233 — $\beta \ne 0$: parabolic asymptote $u^3=A{t}^2$

• FOURTEENTH species — single $u^3=A{t}^2$.

§234 — $\beta =0,\gamma \ne 0$: simultaneous parabolic and rectilinear

The leading equation $\alpha u^3+\gamma tu=0$ factors as $u(\alpha u^2+\gamma t)=0$, yielding both

• a parabolic asymptote $u^2=At$, and

• a straight-line asymptote $u=-ϵ/\gamma$, of the species $u=A/t$, in the direction parallel to the parabola’s axis.

• FIFTEENTH species — one $u^2=At$ together with one $u=A/t$, the latter parallel to the axis of the parabola.

§235 — $\beta =0,\gamma =0,ϵ\ne 0$: $u^3=At$

The leading equation reduces to $\alpha u^3+ϵt=0$, a parabolic asymptote of the species $u^3=At$.

• SIXTEENTH species — single $u^3=At$.

Summary table

Species	Case	Asymptote signature
I	1	one $u=A/t$
II	1	one $u=A/t^2$
III	2	three $u=A/t$
IV	2	two $u=A/t$ + one $u=A/t^2$
V	2	three $u=A/t^2$
VI	3	one $u=A/t$ + one $u^2=At$
VII	3	one $u=A/t^2$ + one $u^2=At$
VIII	3	one $u=A/t$ (double factor gives no branch)
IX	3	one $u=A/t^2$ (double factor gives no branch)
X	3	one $u=A/t$ + two parallel $u=A/t$
XI	3	one $u=A/t^2$ + two parallel $u=A/t$
XII	3	one $u=A/t$ + one $u^2=A/t$
XIII	3	one $u=A/t^2$ + one $u^2=A/t$
XIV	4	one $u^3=A{t}^2$
XV	4	one $u^2=At$ + one $u=A/t$ parallel to its axis
XVI	4	one $u^3=At$

Related pages

• chapter-9-on-the-species-of-third-order-lines
• hyperbolic-and-parabolic-branches — the $u^j=A/t^k$ vs $u^j=A{t}^k$ catalogue.
• double-factor-asymptote-cases — abstract apparatus invoked in Case 3.
• triple-factor-asymptote-cases — abstract apparatus invoked in Case 4.
• simplest-cubic-equations — simplest representative equation for each species, with Newton mapping.
• euler-vs-newton-cubic-species — why 16 vs 72.