determinations-of-a-general-equation

Determinations of the General Equation of Order $n$

Summary: The general polynomial equation of order $n$ carries $\frac{(n+1)(n+2)}{2}$ arbitrary constants (one per monomial $x^i{y}^j$ with $i+j\le n$), but only $\frac{n(n+3)}{2}=\frac{(n+1)(n+2)}{2}-1$ of them are independent. The missing one is absorbed by the projective scaling of the equation: multiplying every constant by a common factor leaves the curve unchanged, so the curve is determined by the ratios of the constants rather than by their absolute values. This is the count that governs how many points are needed to fix a curve — see curve-through-given-points.

Sources: chapter4

Last updated: 2026-04-24

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The observation (§74)

Once all the constants in the general equation of order $n$ are given definite values, the curve is fixed with respect to the chosen axis. Different values pick out different curves from the family. So the general equation of order $n$ is an $m$-parameter family of curves — but what is $m$?

The naive answer from general-equation-of-order-n is $m=\frac{(n+1)(n+2)}{2}$, the number of monomials $x^i{y}^j$ with total degree $i+j\le n$. Euler calls this the number of constants. But the number of determinations — the number of values that can be independently chosen to specify a curve — is one less.

The reduction (§75)

This equation $\alpha +\beta x+\gamma y=0$ would seem to have three different determinations, from the three constants $\alpha ,\beta ,$ and $\gamma$. However, we know from the nature of equations that the equation is really determined by the ratio between these constants, that is, the ratio of two of them to the third. (source: chapter4, §75)

Euler’s example: if $\beta =-\alpha$ and $\gamma =2\alpha$, the equation $\alpha -\alpha x+2\alpha y=0$ is already determined — divide through by $\alpha$ (assuming $\alpha \ne 0$) and it becomes $1-x+2y=0$. The $\alpha$ never had any effect on which curve was described.

The same reduction applies at every order: multiplying every coefficient by a common nonzero scalar does not change the zero set. So one of the $\frac{(n+1)(n+2)}{2}$ constants can be divided out, and only $\frac{(n+1)(n+2)}{2}-1$ independent determinations remain.

The formula

$$\text{Determinations at order }n=\frac{(n+1)(n+2)}{2}-1=\frac{n^2+3n}{2}=\frac{n(n+3)}{2}.$$

Euler writes the first form (§75) and then uses the second in §§76–81 to state how many points a curve is fixed by:

Order $n$	Constants $\frac{(n+1)(n+2)}{2}$	Determinations $\frac{n(n+3)}{2}$
1	3	2
2	6	5
3	10	9
4	15	14
5	21	20
6	28	27
7	36	35

The $\frac{n(n+3)}{2}$ column is also the number of points that pin down the curve (§§78–80). See curve-through-given-points.

Why the reduction is always by exactly one

The equation of an algebraic curve is a homogeneous condition on its coefficients: if $(\alpha ,\beta ,\gamma ,\dots )$ is a valid coefficient tuple for a given curve, so is $(\lambda \alpha ,\lambda \beta ,\lambda \gamma ,\dots )$ for any $\lambda \ne 0$. The family of coefficient tuples per curve is therefore a one-parameter family (a line through the origin in the coefficient space), and a single curve corresponds to a single direction in that space. So the dimension of the space of curves is one less than the number of constants.

In modern language: the space of degree-$n$ curves is the projective space ${\mathbb{P}}^N$ with $N=\frac{(n+1)(n+2)}{2}-1=\frac{n(n+3)}{2}$. Euler does not use this language but does the correct count.

Relation to §58 of chapter 3

Chapter 3’s §58 noted another, compounding effect: even after fixing the $\frac{n(n+3)}{2}$ determinations, different parameter choices can still describe the same curve, because the coordinate system itself is a choice. Different axis/origin/obliquity selections convert one equation into another of the same degree — see coordinate-transformations and general-equation-of-a-curve.

So there are two separate sources of redundancy in the $\frac{(n+1)(n+2)}{2}$ coefficients:

1. Projective scaling (this page, §75): one degree of freedom, always present. Reduces the count to $\frac{n(n+3)}{2}$.
2. Coordinate choice (§58): several more degrees of freedom (origin: 2; axis direction: 1; obliquity: 1 in the oblique case), depending on which coordinate change one admits as an equivalence.

The first reduction is absolute — it is a property of the equation. The second is contextual — it depends on whether one cares to distinguish curves that differ only by a rigid motion or change of coordinates. For the point-fitting problem of §§76–81, the axis is considered fixed (a point’s abscissa and ordinate are defined against that axis), so only the first reduction applies.

Pedagogical note

Euler does not write down the closed form $\frac{n(n+3)}{2}$ explicitly in §75 — he gives the example (3 → 2, 6 → 5, 10 → 9) and then says “In general, the general equation for a line of the $n^{\text{th}}$ order has $\frac{(n+1)(n+2)}{2}-1$ determinations.” In §81 he does write $\frac{n(n+3)}{2}$, deriving it from the algebraic identity $\frac{(n+1)(n+2)}{2}-1=\frac{n(n+3)}{2}$.

Related pages

• chapter-4-on-the-special-properties-of-lines-of-any-order
• general-equation-of-order-n
• curve-through-given-points
• order-of-an-algebraic-curve
• straight-line-equation
• coordinate-transformations
• general-equation-of-a-curve