Chapter 11: On Fourth Order Lines

Summary: Carries the chapter-9 method (classify by branches at infinity) over to the quartic. Eight cases on the linear-factor structure of the principal member produce one hundred forty-six genera in total. Euler enumerates the first 24 in detail and only sketches the bulk of the rest, conceding the work is too long to complete here. Closes with the observation that fifth and higher orders would each demand a whole volume.

Sources: chapter11 (§§260–271).

Last updated: 2026-05-02.

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Program

§260 — the general fourth-order equation has fifteen terms,

$$\alpha y^4+\beta y^{3}x+\gamma y^2{x}^2+\delta y{x}^3+ϵx^4+\zeta y^3+\eta y^{2}x+\theta y{x}^2+\iota x^3+\kappa y^2+\lambda yx+\mu x^2+\nu y+\xi x+o=0.$$

The classification follows the program of chapter-9-on-the-species-of-third-order-lines: rotate so a real linear factor of the principal member becomes a coordinate axis, refine each factor through the chapter-8 machinery, and record what species ($u=A/t,A/t^2,A/t^3$) or parabolic species ($u^2=At,u^2=A/t,u^3=A{t}^2,u^3=At,u^4=A{t}^3,u^4=At$) each branch refines to. The genus is the multiset of these branch species at infinity.

Eight cases

The principal member $\alpha y^4+\beta y^{3}x+\gamma y^2{x}^2+\delta y{x}^3+ϵx^4$ is a homogeneous quartic in $(y,x)$, so over the reals it factors into either zero, two, or four real linear factors, with up to triple- or quadruple-coincidence. §260 lists the eight cases:

Case	Real factors	Coincidences	§§	Genus range	Count
I	none	—	§261	I	1
II	2	distinct	§262	II–VII	6
III	2	equal	§263	VIII–XIV	7
IV	4	all distinct	§264	XV–XXIV	10
V	4	one doubled, two distinct	§265	XXV–LXIV	40
VI	4	two doubled pairs	§266	LXV–CXI	47
VII	4	one tripled, one distinct	§267	CXII–CXXXV	24
VIII	4	all four equal	§§268–270	CXXXVI–CXLVI	11

Total: 146 genera (§270). See fourth-order-genera-enumeration for the per-case derivation, signature lists, and impossibility arguments that prune the naive products.

Detail vs. sketch

Euler is explicit that this chapter cannot be a complete treatment. Cases I–IV (24 genera) are written out. Cases V and VI carry the count to one hundred eleven via a multiplicative argument plus exclusions ($6\times 7-2=40$ for V, $7\times 7-2=47$ for VI), but he declines to enumerate them: “It would take too long to describe all of these here. Since there is not time to treat in detail all of these genera, we cannot say for sure that all of them are real. Whoever wishes, can take it upon himself to perform this task, and if necessary, emend this classification.” (§265). Case VII gets a partial table; Case VIII is enumerated in full.

Closing remark

§271 — having stopped at order 4, Euler observes that “a whole volume were to be dedicated to this one task” if the same enumeration were attempted at order five and beyond. The number of genera grows much faster than the number of cases. The book accordingly turns away from this classification programme; the principal properties of higher-order lines, Euler notes, can be derived in the same way as for cubics (chapter-10-on-the-principal-properties-of-third-order-lines) when needed.

Related pages

• chapter-9-on-the-species-of-third-order-lines — the cubic analogue (16 species), which sets the template.
• chapter-7-on-the-investigation-of-branches-which-go-to-infinity — the criterion that links real linear factors of the principal member to infinite branches.
• chapter-8-concerning-asymptotes — the refinement machinery whose outputs ($u=A/t^k$, $u^j=A{t}^k$, etc.) are what Euler tabulates here.
• hyperbolic-and-parabolic-branches — the §218 catalogue from which each species label is drawn.
• fourth-order-genera-enumeration — full per-case derivation of the 146 genera.