Substitution

Summary: Euler’s second transformation technique: rather than rewrite $y=f(z)$ with the same variables, introduce a new variable $x$ such that both $y$ and $z$ become functions of $x$. Used to remove radicals and to make implicit relations explicit.

Sources: chapter3

Last updated: 2026-04-23

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The idea

Chapter 2 transforms a function by rewriting it with the same variable $z$. Chapter 3 transforms differently: introduce a third variable $x$ such that both the independent variable $z$ and the dependent variable $y$ are defined as functions of $x$ (source: chapter3, §46).

Setting any value of $x$ then simultaneously determines values of $z$ and $y$ that satisfy the original relation. The resulting pair $(z(x),y(x))$ is what we would today call a parametrization.

Euler’s opening example: given $y=\frac{1-z^2}{1+z^2}$, let $z=\frac{1-x}{1+x}$. Substituting yields $y=\frac{2x}{1+x^2}$. At $x=\frac{1}{2}$: $z=\frac{1}{3}$ and $y=\frac{4}{5}$, consistent with $y=(1-z^2)/(1+z^2)$ at $z=\frac{1}{3}$ (source: chapter3, §46).

Why substitute

Euler gives two motivations (source: chapter3, §46):

1. Remove radicals. If $y$ as a function of $z$ contains a radical, a well-chosen substitution can express both $z$ and $y$ as rational functions of $x$. See rationalizing-substitutions.
2. Handle implicit relations. If $y$ and $z$ are tied by an implicit equation of higher degree, so that neither can be solved for explicitly in terms of the other, a substitution may yield explicit formulas for both in terms of $x$. See homogeneous-substitution.

Both motivations point to what modern language calls a rational parametrization of an algebraic curve.

Position in the chapter

• §46 — motivation and the opening example.
• §47–§51 — removing radicals (see rationalizing-substitutions).
• §52–§58 — implicit relations, using $y=xz$ or $y=x^m{z}^n$ (see homogeneous-substitution).

Related pages

• chapter-3-on-the-transformation-of-functions-by-substitution
• rationalizing-substitutions
• homogeneous-substitution
• rational-parametrization-of-the-circle
• folium-of-descartes
• chapter-2-on-the-transformation-of-functions — the first kind of transformation.