degree-invariance

Degree Invariance under Change of Coordinates

Summary: The degree of the polynomial equation of an algebraic curve is unchanged by any rigid coordinate transformation — translation, rotation, axis-swap, or their composition — and also by a change to oblique coordinates. Since the substitutions involved are all first-degree in the new coordinates, no substitution can raise or lower the total degree. Consequently, two equations of different degrees cannot represent the same curve.

Sources: chapter2

Last updated: 2026-04-24

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The statement (§§37–38)

If the degrees of the two equations are different, that is, if the highest power of $x$ and $y$ is different from the highest power of $t$ and $u$, then the curves indicated by the two equations are certainly different. This follows from the fact that the degree of an equation in $x$ and $y$ is the same as the degree of the equation which results when we make the substitutions $x=mu+nt-f$ and $y=nu-mt-g$. (source: chapter2, §37)

Restated: the total degree — the maximum of $i+j$ over monomials $x^i{y}^j$ in the equation — is an invariant of the curve, not a feature of the particular coordinate system chosen.

Why it holds

The substitutions for a general rigid motion are

$$x=mu+nt-f,y=nu-mt-g,$$

each of which is first-degree in $t,u$. Replacing $x$ by a first-degree expression in $t,u$ keeps the degree of any monomial $x^i{y}^j$ at $i+j$; it cannot be raised (at most first-degree goes in, and the product of a degree-$k$ polynomial with a degree-1 is degree $k+1$ — matching $x^i$ still yielding degree $i$ after full substitution), and it cannot be lowered except by algebraic cancellation that would require the original equation to have had a distinguished higher-degree term that cancels. Euler takes the “cannot be lowered” direction as read: the degree after substitution equals the degree before.

In §46 the same argument is repeated for the oblique case: the general equation with oblique coordinates is again obtained by a first-degree substitution in new coordinates $r,s$, “so it is clear that the most general equation is of the same degree as the original equation in $x$ and $y$” (source: chapter2, §46). Hence degree is invariant under all transformations considered in the chapter, rectangular or oblique.

Usage: a cheap non-equivalence test (§38)

Running the full substitution test of general-equation-of-a-curve is “rather tedious.” The degree check is instant and it often decides the question:

Unless two equations, the one in $x$ and $y$, and the other in $t$ and $u$, both have the same degree, we can immediately conclude that the curves expressed by the equations are also different. The only time when there is cause for doubt is when they both have the same degree. (source: chapter2, §38)

So the practical workflow is:

1. Compare the degrees. If they differ, the curves are different.
2. If the degrees agree, only then is the substitution test (§36) worth running.

For large degrees Euler promises “a more expeditious method” below (§38), foreshadowing later chapters.

Implications for the classification of curves

Degree is therefore a well-defined attribute of an algebraic curve, independent of the axis, origin, or obliquity of the coordinates. This underwrites the classical classification of algebraic curves by order:

• order 1: straight lines (all first-degree equations — see straight-line-equation);
• order 2: conic sections (Book II treats these starting with chapter 3, chapter3);
• order 3 and higher: cubics, quartics, …

Without the §37 / §46 invariance theorem, the order of a curve would depend on the coordinate choice, and the classification would be meaningless.

Related pages

• chapter-2-on-the-change-of-coordinates
• coordinate-transformations
• general-equation-of-a-curve
• straight-line-equation
• oblique-coordinates
• algebraic-and-transcendental-curves