General Equation of Order $n$

Summary: The general polynomial equation of order $n$ in two coordinates $x,y$ is the sum of all monomials $x^i{y}^j$ with $i+j\le n$, each with its own arbitrary constant. The number of such monomials — and hence the number of arbitrary constants — is the $n$-th triangular number $\frac{(n+1)(n+2)}{2}$. This gives a fixed template for the classification of algebraic curves by order-of-an-algebraic-curve.

Sources: chapter3

Last updated: 2026-04-24

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Order by order (§§52–57)

Order 1 (§52) — 3 terms:

$$\alpha +\beta x+\gamma y=0.$$

The only species is the straight line (see straight-line-equation).

Order 2 (§§53–54) — 6 terms:

$$\alpha +\beta x+\gamma y+\delta x^2+ϵxy+\zeta y^2=0.$$

These are the conic sections: circle, ellipse, parabola, hyperbola. “We will deduce all of these from the general equation in a subsequent chapter” (source: chapter3, §54).

Order 3 (§55) — 10 terms:

$$\begin{array}{rl}\alpha +\beta x+\gamma y& +\delta x^2+ϵxy+\zeta y^2\\ +& \eta x^3+\theta x^{2}y+\iota x{y}^2+\kappa y^3=0.\end{array}$$

Euler attributes the classification of the species at this order to Newton. Since this equation already has more arbitrary constants than the second-order equation, it “contains many more species.”

Order 4 (§56) — 15 terms:

$$\begin{array}{rl}\alpha & +\beta x+\gamma y+\delta x^2+ϵxy+\zeta y^2\\ & +\eta x^3+\theta x^{2}y+\iota x{y}^2+\kappa y^3\\ & +\lambda x^4+\mu x^{3}y+\nu x^2{y}^2+\xi x{y}^3+o{y}^4=0.\end{array}$$

“Many more species in this order than in the preceding.”

Order 5 (§57) adds the six degree-5 monomials $x^5,x^{4}y,x^3{y}^2,x^2{y}^3,x{y}^4,y^5$, for 21 terms in total. Order 6 has 28 terms, and so on by triangular numbers.

The count: $\frac{(n+1)(n+2)}{2}$

The general equation for the lines of order $n$ has $\frac{(n+1)(n+2)}{2}$ terms, and the same number of arbitrary constants. (source: chapter3, §57)

The reason: the monomials of total degree exactly $k$ are $x^k,x^{k-1}y,\dots ,x{y}^{k-1},y^k$ — that’s $k+1$ of them. Summing $k=0,1,\dots ,n$ gives

$$1+2+3+\cdots +(n+1)=\frac{(n+1)(n+2)}{2}.$$

Written out: 3, 6, 10, 15, 21, 28, 36, 45, 55, … — the triangular numbers.

Caveats on counting species

The term-count is the number of monomials, not the number of genuinely distinct curves of order $n$. Two obstacles stand between “arbitrary-constant count” and “species count”:

• Coordinate redundancy (§58). Chapter 2 showed that a single curve has infinitely many equations, one per choice of axis/origin/obliquity. So varying the $\frac{(n+1)(n+2)}{2}$ constants does not always produce a new curve: “although the equations of the same degree may be different, the curves may be the same” (source: chapter3, §58). Some of the constants are “used up” by the coordinate system rather than by the curve.
• Reducibility (§61). If the polynomial factors, the equation does not describe a single curve at all, but a bundle of simpler curves. See complex-curves.

Both effects are why naming and counting species at each order is a non-trivial geometric task, not just an algebraic one. Order 2 is easy (four species). Order 3 was done by Newton. Order 4 and higher are substantially harder.

Rectangular vs. oblique — no gain (§§54, 55)

For second and higher orders Euler takes the coordinates to be rectangular, because “obliquity of the ordinate brings no more generality to the equation” (source: chapter3, §55, §56). That is: the oblique general equation has the same number of free parameters as the rectangular general equation, so permitting obliquity would not enlarge the family of curves — it would only introduce a redundant parameter. This is a concrete payoff of oblique-coordinates’s §45–46 analysis.

The terminology shift Euler rejects (§56)

A rival convention renames each order downward by one: since order 1 contains no curved lines, it is not counted, and order 2 is then called “first-order curves,” order 3 “second-order curves,” and so on. Euler notes this usage (“Lines of order four are frequently called third order curves, since lines of order two are reputed to be first order curves”) but keeps his own numbering. A reader crossing into later literature (Newton’s cubics, in particular) should check which convention is in force.

Constants vs. determinations (chapter 4, §75)

The count $\frac{(n+1)(n+2)}{2}$ is the number of constants, but one constant is absorbed by projective scaling (the equation is fixed only up to a common scalar). Effective determinations are therefore

$$\frac{(n+1)(n+2)}{2}-1=\frac{n(n+3)}{2},$$

which is also the number of points needed to pin down a curve of order $n$. See determinations-of-a-general-equation and curve-through-given-points.

Related pages

• chapter-3-on-the-classification-of-algebraic-curves-by-orders
• chapter-4-on-the-special-properties-of-lines-of-any-order
• determinations-of-a-general-equation
• curve-through-given-points
• order-of-an-algebraic-curve
• complex-curves
• straight-line-equation
• oblique-coordinates
• degree-invariance