Chapter 2: On the Change of Coordinates

Summary: Euler works out the full apparatus of planar coordinate transformations — translation, rotation, axis-swap, and their composition, plus the extension to oblique ordinates. The payoff is a pair of structural results: every curve has a single general equation containing four (or five, with obliquity) free parameters that specialize to every possible coordinate system, and the degree of that equation is invariant under all transformations, so different degrees mean different curves.

Sources: chapter2

Last updated: 2026-04-24

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Why this chapter

Chapter 1 defined the coordinate apparatus but left two arbitrary choices — axis and origin — embedded in the equation of every curve. As a result, one curve has infinitely many equations, and it is not obvious when two equations describe the same curve. Chapter 2 answers both halves of that question: it gives the substitutions that generate all equations of a fixed curve, and it gives a decision procedure (plus a cheap necessary test) for equivalence.

Since by a change of axis or origin innumerable different equations express the nature of the same curve, all of these equations should be compared, so that when one is given, all of the others can be found. (source: chapter2, §24)

Structure of the chapter

§§23–24 motivate the study. Different choices of axis $RS$ and origin $A$ yield different equations for the same curve; the reverse is not true.

§§25–28 — translation. Moving the origin along the axis substitutes $x\mapsto t-f$. Moving to a parallel axis additionally substitutes $y\mapsto u-g$. See coordinate-transformations.

§§29–30 — axis swap. A new axis perpendicular to the old at the common origin simply exchanges the roles of abscissa and ordinate. Euler concludes that the two coordinates are on equal footing: “making no distinction which is called the abscissa or ordinate” (source: chapter2, §29). Sign-direction conventions for positive abscissas and positive ordinates are likewise free.

§§31–32 — rotation. A rotation by angle $q$ (with $\sin q=m$, $\cos q=n$, so $m^2+n^2=1$) about the common origin uses the substitutions $x=mu+nt$, $y=nu-mt$.

§§33–34 — general rigid motion. Composing translation and rotation yields the master substitution

$$x=mu+nt-f,y=nu-mt-g,m^2+n^2=1,$$

parametrized by the four free quantities $f,g,m,n$.

§35 — the general equation of a curve. Since every orthogonal axis in the plane is captured by a choice of $(f,g,m,n)$, the equation obtained from the master substitution contains, in a single expression, every orthogonal equation the curve admits. See general-equation-of-a-curve.

§36 — decision procedure. Given two equations, test whether they represent the same curve by substituting the master transformation into one and trying to match the other. Euler runs the test on the worked example $y^2-ax=0$ vs. $16u^2-24tu+9t^2-55au+10at=0$, finding $m=3/5$, $n=4/5$, $g=a$, $f=-a$: the curves coincide.

§§37–38 — degree invariance. Because the master substitution is first-degree in the new coordinates, it preserves the total degree of the equation. In particular, equations of different degrees cannot be equivalent — an instant test that often short-circuits the tedious §36 procedure. See degree-invariance.

§§39–41 — straight lines. A line parallel to the axis has the rectangular equation $y=a$; under the master substitution this becomes $\alpha t+\beta u-b=0$, so every first-degree equation in any orthogonal coordinates represents a straight line, and vice versa. Degenerate cases ($\alpha =0$ or $\beta =0$) are handled directly. See straight-line-equation.

§§42–46 — oblique coordinates. The ordinate is allowed to make any fixed angle $\varphi$ with the axis. The earlier apparatus extends: rectangular-to-oblique (§43) and oblique-to-rectangular (§44) substitutions, culminating (§45) in the most general equation of a curve, now with five free parameters (axis position, origin, obliquity). The degree is still preserved (§46). See oblique-coordinates.

Notable points

• The equivalence test of §36 is one-sided: it works by substituting into one equation and matching the other, so one has to choose which side to transform. In the worked example Euler multiplies both equations by scaling constants ($n^2$ and $16$) to align leading coefficients — this is the computational bottleneck.
• Degree invariance is the decisive shortcut. Two equations of different degrees represent different curves, always; no computation needed. The substitution test is only required when the degrees match (§38).
• The chapter makes the abscissa and ordinate interchangeable (§29): the curve does not care which axis is called “horizontal.” This is Euler’s way of anticipating what we now phrase as “the equation is symmetric under swapping the two coordinates up to relabeling.”
• The parameters $m,n$ are not independent — $m^2+n^2=1$ — so the orthogonal master substitution has three free parameters (one rotation angle $q$ plus two translations $f,g$), and the oblique version has four (adding $\varphi$). This is the modern “group of rigid motions” count before Lie groups existed.

What this buys for the rest of Book II

• The order of an algebraic curve is well-defined. Conic sections can be defined as “curves of order 2” without having to fix a coordinate system. This underwrites chapters 3 and following, which begin the systematic study of second-degree curves.
• Normal forms become meaningful. Given a curve of a certain degree, one can seek the special axis and origin in which the equation is simplest — a programme Euler carries out for conics in chapter 3, for cubics later, and so on.
• Straight lines are characterized purely algebraically. This bridges synthetic and analytic geometry in one line: linear equation $\iff$ straight line.

Related pages

• coordinate-transformations
• general-equation-of-a-curve
• degree-invariance
• straight-line-equation
• oblique-coordinates
• chapter-1-on-curves-in-general