thread-constants

Thread Constants and Equational Homogeneity

Summary: Euler’s term for the constants $a,b,c,\dots$ that appear in a curve’s equation alongside the variables $x,y$. Every term of the equation must have the same thread degree — the total count of variable and constant factors — because otherwise we would be comparing heterogeneous magnitudes. The same homogeneity that governs purely-variable terms (degree in $x$ and $y$) governs the joint dependence on variables and constants.

Sources: chapter18 §435.

Last updated: 2026-05-12.

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The dimensional rule (§435)

A curve equation looks like $F(x,y;a,b,c,\dots )=0$ with constants $a,b,c,\dots$ that Euler calls thread constants. Take any term: it is a product of some number of threads, say $n$. Euler’s rule: in every other term of the same equation the number of threads multiplied must also be $n$, so that we add only homogeneous quantities. If one term were a product of two threads and another of three, we would be comparing a quantity of dimension $L^2$ with one of dimension $L^3$, which is meaningless.

Exception (§435): a thread constant that happens to equal $1$ or some pure number (without geometric magnitude) does not count. The dimensional accounting only constrains constants that carry the same “linear” thread-dimension as $x$ and $y$.

Consequence for purely-variable equations

If no thread constants appear, the rule degenerates: every term must have the same total degree in $x$ and $y$ alone, so the equation is homogeneous in $x,y$. Euler had already noted in earlier chapters that such an equation does not give a curve — it gives a bundle of straight lines all passing through the origin (the only common solution of $x=0,y=0$ is degenerate, otherwise the equation factors into linear forms over $y/x$).

Why this matters for chapter 18

Thread homogeneity is the algebraic backbone of the chapter’s three classification statements:

• Similar curves (similar-curves): one thread constant $a$ + variables $x,y$, all three appearing in the same degree per term, means substituting $a\mapsto a/n,x\mapsto x/n,y\mapsto y/n$ leaves the equation unchanged after clearing a common factor of $n^{\text{thread-degree}}$.
• Affine curves (affine-curves): if $x$ and $y$ are scaled independently ($x\mapsto X/m$, $y\mapsto Y/n$), thread homogeneity is broken, but a residual zero-degree condition in $(X,m)$ for the ratio $Y/n$ persists.
• Infinite congruent copies (infinite-copies-of-curve): a varying constant $a$ that is not homogeneous with the variables can encode position rather than scale.

Reading

Thread-degree is to chapter 18 what order is to chapter 3: a single invariant that organizes the rest of the chapter. But where order counts the maximum degree in $x,y$, thread-degree counts the total degree in variables and constants together, and is constant across the whole equation rather than maximized over it.

Related pages

• similar-curves
• affine-curves
• infinite-copies-of-curve
• order-of-an-algebraic-curve
• general-equation-of-order-n