Chapter 3: On the Transformation of Functions by Substitution

Summary: Euler introduces the second kind of transformation promised in Chapter 2: replacing the variable $z$ with a new variable $x$ in terms of which both $z$ and $y$ are defined. Two applications dominate the chapter — removing radicals, and making implicit algebraic relations explicit.

Sources: chapter3

Last updated: 2026-04-23

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Overview

Where Chapter 2 kept the same variable and rewrote the function, Chapter 3 introduces a fresh variable $x$ and simultaneously defines both $y$ and $z$ as functions of it (source: chapter3, §46). In modern terms, Euler is constructing rational — or at least radical-free — parametrizations of algebraic relations.

The chapter organizes around two large themes:

1. Removing radicals (§47–§51). See rationalizing-substitutions.
2. Parametrizing implicit polynomial relations (§52–§58). See homogeneous-substitution.

The generic tool used in (1) is a tailored substitution that exploits the form of the radical; in (2) it is the homogeneous ansatz $y=xz$ (or the more general $y=x^m{z}^n$).

Structure of the chapter

§46 — What substitution means

If $y$ is a function of $z$, we may introduce a new variable $x$ by specifying $z=g(x)$; both $y$ and $z$ then become functions of $x$. The method is justified either because it removes a radical from $y(z)$ or because it makes an implicit higher-degree relationship between $y$ and $z$ explicitly solvable (source: chapter3, §46). The opening example $y=(1-z^2)/(1+z^2)$ with $z=(1-x)/(1+x)$ is the rational-parametrization-of-the-circle (with $a=1$).

See substitution.

§47–§51 — Removing radicals

A catalog of substitutions keyed to the form of the radical:

• §47: $y=\sqrt{a+bz}$ — set $y=bx$.
• §48: $y=(a+bz{)}^{m/n}$ — set $y=x^m$.
• §49: $y={\left(\frac{a+bz}{f+gz}\right)}^{m/n}$ — set $y=x^m$.
• §50: $y=\sqrt{(a+bz)(c+dz)}$ — set the radical equal to $(a+bz)x$. Special case $y=\sqrt{a^2-z^2}$ is the circle (source: chapter3, §47–§50).
• §51: $y=\sqrt{p+qz+r{z}^2}$ — cases I, II, III on the signs of $p$ and $r$; case III reduces to §50 when the quadratic has real factors, or is otherwise handled by a variant ansatz (source: chapter3, §51).

Euler acknowledges that other forms of radical relation exist and “cannot be reduced to a form without radicals by a substitution without radicals” (source: chapter3, §51). See rationalizing-substitutions.

§52 — Three-term relations

$a{y}^{\alpha }+b{z}^{\beta }+c{y}^{\gamma }{z}^{\delta }=0.$

Substituting $y=x^m{z}^n$ gives three ways to choose $n$ so two exponents of $z$ match and the common factor of $z$ can be stripped. The worked example is $y^3+z^3-cyz=0$, the folium-of-descartes, whose Method-I form gives the classical rational parametrization $z=cx/(1+x^3)$, $y=c{x}^2/(1+x^3)$ (source: chapter3, §52).

§53 — A posteriori construction

Given a rational parametrization $z(x),y(x)$ one can run §52 backwards and reconstruct an implicit polynomial relation $F(y,z)=0$ (source: chapter3, §53).

§54–§57 — Exactly two total degrees

When the monomials of the implicit relation involve exactly two total degrees $m>n$, $y=xz$ followed by division by $z^n$ leaves a single power $z^{m-n}$ equal to a rational function of $x$. Particular cases worked out:

• §54: $a{y}^2+byz+c{z}^2+dy+ez=0$ (degrees 2 and 1).
• §55: degrees 3 and 2.
• §56: degrees 2 and 0 (i.e. $a{y}^2+byz+c{z}^2=d$).
• §57: general degrees $m,n$.

All reduce to $z={\left(\text{rational in }x\right)}^{1/(m-n)}$, $y=xz$ (source: chapter3, §54–§57).

§58 — Three total degrees in arithmetic progression

If the implicit relation involves exactly three total degrees forming an arithmetic progression, $y=xz$ produces an equation quadratic in a power of $z$, solvable via the quadratic formula. Euler gives three examples — a cubic-quadratic-linear relation, a sextic with degrees $5,3,1$, and a relation with degrees $10,7,4$ (source: chapter3, §58 examples I–III).

See homogeneous-substitution.

Notable points

• The chapter is remarkable for quietly discovering, case by case, that certain algebraic curves admit rational parametrizations. Modern language: these are precisely the genus-zero cases. Euler does not yet have the language, but the phenomenon is correctly identified.
• §46 and §50 together recover the classical half-angle parametrization of the circle.
• The §52 Method-I formula applied to $y^3+z^3=cyz$ gives the standard parametrization of the folium of Descartes.
• The §58 trick — reduce three AP-spaced degrees to a quadratic in $z^k$ — is an early instance of what would later be called “weighted homogeneity.”
• At the end of §51 Euler explicitly admits the limits of algebraic substitution: “other cases, which are not discussed in this treatise, cannot be reduced to a form without radicals by a substitution without radicals” (source: chapter3, §51). This is a frank recognition that not every algebraic relation is rational.

Why this chapter matters

Together with Chapter 2, this chapter completes Euler’s algebraic toolkit for simplifying functions. Chapter 2 rewrote; Chapter 3 reparametrizes. Both are set-up: once an expression has been broken into simple pieces and/or written rationally in a convenient variable, later chapters can integrate, sum, and expand it as a series.

Related pages

• substitution
• rationalizing-substitutions
• homogeneous-substitution
• rational-parametrization-of-the-circle
• folium-of-descartes
• chapter-2-on-the-transformation-of-functions