Simplest Cubic Equations

Summary: Euler’s §237 catalogue of the simplest representative equation for each of the sixteen cubic species, together with the Newtonian species numbers each one subsumes.

Sources: chapter9 (§237).

Last updated: 2026-04-30.

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For each species Euler exhibits a generic equation in its simplest form, with constraints chosen so that the form lies cleanly inside that species and not on the boundary with a neighbouring one. The Newtonian numbers (1–72) record the partition of Newton’s classification under Euler’s coarser scheme.

First case

FIRST (single asymptote $u=A/t$):

$y(x^2-2mxy+n^2{y}^2)+a{y}^2+bx+cy+d=0,m^2>n^2,b\ne 0.$

Newton’s species 33–38.

SECOND (single asymptote $u=A/t^2$):

$y(x^2-2mxy+n^2{y}^2)+a{y}^2+cy+d=0,m^2<n^2.$

Newton’s species 39–45.

Three distinct factors

THIRD (three asymptotes $u=A/t$):

$y(x-my)(x-ny)+a{y}^2+bx+cy+d=0,$

with $b\ne 0$, $mb+c+a^2/(m-n{)}^2\ne 0$, $nb+c+a^2/(m-n{)}^2\ne 0$, $m\ne n$. Newton’s species 1–9, plus 24, 25, 26, 27 if $a=0$.

FOURTH (two of $u=A/t$, one of $u=A/t^2$):

$y(x-my)(x-ny)+a{y}^2+cy+d=0,$

with $c+a^2/(m-n{)}^2\ne 0$, $m\ne n$. Newton’s species 10–21, plus 28, 29, 30, 31 if $a=0$.

FIFTH (three of $u=A/t^2$):

$y(x-my)(x-ny)+a{y}^2-\frac{a^{2}y}{(m-n{)}^2}+d=0,m\ne n.$

Newton’s species 22, 23, 32.

Double factor

SIXTH ($u=A/t$ and $u^2=At$):

$y^2(x-my)+a{x}^2+bx+cy+d=0,a\ne 0,2m^3{a}^2-mb-c\ne 0.$

Newton’s species 46–52.

SEVENTH ($u=A/t^2$ and $u^2=At$):

$y^2(x-my)+a{x}^2+bx+m(2m^2{a}^2-b)y+d=0,a\ne 0.$

Newton’s species 53–56.

EIGHTH ($u=A/t$, double factor non-asymptotic):

$y^2(x-my)+b^{2}x+cy+d=0,c\ne -m{b}^2,b\ne 0.$

Newton’s species 61, 62.

NINTH ($u=A/t^2$, double factor non-asymptotic):

$y^2(x-my)+b^{2}x-m{b}^{2}y+d=0,b\ne 0.$

Newton’s species 63.

TENTH ($u=A/t$ + two parallel $u=A/t$):

$y^2(x-my)-b^{2}y+cy+d=0,c\ne m{b}^2,b\ne 0.$

Newton’s species 57, 58, 59. (As printed; the parallel structure with the EIGHTH and ELEVENTH forms suggests the leading lower-order term is intended to be $-b^{2}x$ rather than $-b^{2}y$ — verify against the source if reusing the formula.)

ELEVENTH ($u=A/t^2$ + two parallel $u=A/t$):

$y^2(x-my)-b^{2}x+m{b}^{2}y+d=0,b\ne 0.$

Newton’s species 60.

TWELFTH ($u=A/t$ and $u^2=A/t$):

$y^2(x-my)+cy+d=0,c\ne 0.$

Newton’s species 64.

THIRTEENTH ($u=A/t^2$ and $u^2=A/t$):

$y^2(x-my)+d=0.$

Newton’s species 65.

Triple factor

FOURTEENTH (single $u^3=A{t}^2$):

$y^3+a{x}^2+bxy+cy+d=0,a\ne 0.$

Newton’s species 67–71.

FIFTEENTH ($u^2=At$ + parallel $u=A/t$):

$y^3+bxy+cx+d=0,b\ne 0.$

Newton’s species 66.

SIXTEENTH (single $u^3=At$):

$y^3+ay+bx=0,b\ne 0.$

Newton’s species 72.

Mapping summary

The 72 Newtonian species partition without overlap into the 16 Eulerian classes:

Newton numbers	Euler species
1–9 (and 24–27 with $a=0$)	III
10–21 (and 28–31 with $a=0$)	IV
22, 23, 32	V
33–38	I
39–45	II
46–52	VI
53–56	VII
57–59	X
60	XI
61, 62	VIII
63	IX
64	XII
65	XIII
66	XV
67–71	XIV
72	XVI

Related pages

• chapter-9-on-the-species-of-third-order-lines
• cubic-species-classification — derivation of the 16 species from the principal-member factor structure.
• euler-vs-newton-cubic-species — why Newton’s count is finer.