straight-line-equation

The Equation of a Straight Line

Summary: Euler proves that every first-degree equation in two orthogonal coordinates, $\alpha x+\beta y=a$, represents a straight line — and that conversely every straight line has such an equation. The proof runs the general-equation machinery of chapter 2 backwards: start from the simplest case $y=a$ (a line parallel to the axis), transform by the general rigid motion, and find that every transformed version remains first-degree.

Sources: chapter2 (figures 12–13)

Last updated: 2026-04-24

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The simplest case (§39)

A straight line $LM$ parallel to the axis $RS$ at perpendicular distance $a$ has equation

$$y=a,$$

since every ordinate has the same length (figure 12).

Every straight line has a first-degree equation (§39)

Apply the general rigid motion (§§33–34 of coordinate-transformations): pick any axis $rs$, with $DG=g$, angle $ODs$ having sine $m$ and cosine $n$. The new abscissa $DQ=t$ and new ordinate $QM=u$ are related to $y$ by

$$y=nu-mt-g,$$

so the condition $y=a$ becomes

$$nu-mt-g-a=0.$$

Multiplying through by a constant $k$, set $\alpha =nk$, $-\beta =-mk$ (i.e. $\beta =mk$), $b=-(g+a)k$. The equation takes the form

$$\alpha u+\beta t+b=0,$$

which is the general first-degree equation in the coordinates $t,u$ of the new axis. “It is clear that every first degree equation in two coordinates exhibits not a curve but a straight line” (source: chapter2, §39).

Every first-degree equation is a straight line (§40)

Conversely, given $\alpha x+\beta y=a$ in orthogonal coordinates $x,y$ relative to some axis $RS$, read off two points on the locus and use similar triangles to recover the line (figure 13):

• setting $y=0$: the locus crosses the axis at $AC=a/\alpha$,
• setting $x=0$: the locus reaches ordinate $AB=a/\beta$ at the origin.

The two points $B$ and $C$ determine a unique straight line. For any third point $M$ on this line, the similar triangles $CPM$ and $CAB$ give $CP:PM=CA:AB$, i.e.

$$\frac{a}{\alpha }-x:y=\frac{a}{\alpha }:\frac{a}{\beta },$$

which rearranges to $\alpha x+\beta y=a$. Every such $M$ satisfies the equation, and the argument is reversible, so the locus is exactly the straight line $BC$.

Degenerate cases (§41)

The general argument breaks down when $\alpha$ or $\beta$ vanishes, but each degenerate case is geometrically clear:

Equation	Line
$\beta y=a$, i.e. $y$ constant	parallel to the axis at distance $a/\beta$
$y=0$	the axis itself
$\alpha x=a$, i.e. $x$ constant	perpendicular to the axis at distance $a/\alpha$ from the origin

In the last case the ordinate is no longer variable — the abscissa takes a single value and every ordinate corresponds to it.

Relation to degree invariance

The result is a corollary of the degree-invariance theorem (§§37, 46 — see degree-invariance). A straight line, being definable in some coordinate system by a first-degree equation ($y=a$), has a first-degree equation in every coordinate system, and no curve with a higher-degree equation can be a straight line. This is why “first-degree equation” and “straight line” are synonymous, regardless of the coordinate system in which the equation is written.

Figures

Figures 11–14

Related pages

• chapter-2-on-the-change-of-coordinates
• coordinate-transformations
• degree-invariance
• general-equation-of-a-curve
• oblique-coordinates