Order of an Algebraic Curve

Summary: The order of an algebraic curve is the degree of any polynomial equation that represents it, once radicals and fractions have been cleared. Because degree-invariance preserves degree under every change of axis, origin, and obliquity, the order is a genuine attribute of the curve — independent of how the coordinate system is laid down. Two rival classifying qualities Euler tries first (the genus of §48 and the term-count of §50) both fail this invariance test and are rejected.

Sources: chapter3

Last updated: 2026-04-24

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The definition (§§51–52)

Since for the same curve, whatever axis and origin is chosen or however the inclination of the coordinates may be varied, the degree of the equation always remains the same, so that the same curve is not referred to different classes. Hence the classifying property is the degree of the equation in the coordinates, whether they are rectangular or oblique. (source: chapter3, §51)

A curve whose equation has degree $n$ is called a line of order $n$. Euler prefers to call all algebraic curves “lines” — then the straight line is no longer a terminological exception (“lines of the first order contain no curved lines, but only straight lines,” source: chapter3, §52).

Why genus fails (§§48–49)

The first guess for a classifying principle is the number of values of $y$ per $x$ — single-valued curves are “first genus,” two-valued “second genus,” and so on. This classification is natural because multi-valued-curves is already one of chapter 1’s main topics.

Euler rejects it with one counter-example. The curve

$$a^{3}y=a^2{x}^2-x^4$$

gives $y$ as a single-valued polynomial in $x$, so it is first genus. Interchange the two coordinates — a legitimate coordinate change per §29 — and the same curve is described by

$$y^4-a^2{y}^2+a^{3}x=0,$$

which is fourth genus. One curve, two different genera: “the genus of the curve is ambiguous, and this is not allowable” (source: chapter3, §49).

Why term-count fails (§50)

The second guess: count the number of terms in the equation (2 terms = first class, 3 terms = second class, …). Euler rejects this with the §36 worked example from chapter 2. The parabola $y^2-ax=0$ has 2 terms, but after a suitable change of axis and origin the same parabola satisfies

$$16u^2-24tu+9t^2-55au+10at=0,$$

which has 5 terms. A different axis would give 3 or 4 terms instead (source: chapter3, §50).

Why degree wins (§51)

Degree is invariant under every coordinate change in coordinate-transformations, rectangular or oblique (see degree-invariance). So whatever axis, origin, or obliquity the student picks, the degree is the same — the curve falls in the same class.

Degree also has the complementary virtue of being decidable from a general equation: whether Euler writes the general second-degree equation with its 6 arbitrary constants all distinct or with most set to zero, the order is still 2. A curve does not “escape” its order by having a sparse equation (§51).

How to read off the order of a given equation (§59)

Given any algebraic equation in $x,y$:

1. Remove irrationality. $y^2=x\sqrt{a^2-x^2}$ squares to $y^4=x^2(a^2-x^2)$.
2. Remove fractions. $y=\frac{a^3-a{x}^2}{a^2+x^2}$ clears to $(a^2+x^2)y=a^3-a{x}^2$.
3. Read off the highest total degree $i+j$ over all monomials $x^i{y}^j$.

Worked examples (source: chapter3, §59):

• $y^2-ax=0$ → order 2.
• $y^2=x\sqrt{a^2-x^2}$ → order 4 (after squaring).
• $y=(a^3-a{x}^2)/(a^2+x^2)$ → order 3 (after clearing the fraction).

The order is total degree, not the degree in either variable separately: $y^2-ax$ has degree 2 in $y$ and 1 in $x$, and the order is 2 (the max over monomials of $i+j$).

Order is stable even when the curve is not (§60)

The equation $y^2=a^2-x^2$ describes a circle in rectangular coordinates and an ellipse in oblique coordinates. Different curves — but both are order 2, because the degree does not change with obliquity. Hence:

All of these different curves belong to the same order, since when the coordinates are changed from oblique to rectangular the order of the curve remains the same. (source: chapter3, §60)

So “order” is coarser than “species”: each order contains multiple species (e.g., order 2 contains the four conic sections), but the order itself is fixed.

Caveat: reducibility (§61)

The construction above assumes the equation is irreducible over the rationals. If the polynomial factors, the equation bundles several curves, and the curve is not properly of its order. See complex-curves.

Geometric shadow of order (chapter 4)

The algebraic order admits a geometric diagnostic: a curve of order $n$ meets any straight line in at most $n$ points, with equality on a generic line. See line-curve-intersection-bound. The bound gives an order lower bound from any intersection count, but not a tight upper bound (a 2-intersection curve could be anything of order $\ge 2$).

Related pages

• chapter-3-on-the-classification-of-algebraic-curves-by-orders
• chapter-4-on-the-special-properties-of-lines-of-any-order
• line-curve-intersection-bound
• degree-invariance
• general-equation-of-order-n
• complex-curves
• multi-valued-curves
• coordinate-transformations
• straight-line-equation