Chapter 1: On Functions in General

Summary: Euler’s opening chapter of the Introductio lays the foundation for the entire work by defining constants, variables, and functions, and classifying functions by their method of construction.

Sources: chapter1

Last updated: 2026-04-23

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Overview

This chapter defines the basic vocabulary of analysis as Euler conceives it. He begins from the distinction between variable-and-constant quantities, gives his famous definition of a function as an “analytic expression,” and then builds a taxonomy that organizes functions by the operations used to construct them. He ends with three structural topics: single-valued-and-multi-valued-functions, even-and-odd-functions, and similar-functions.

Structure of the chapter

Paragraphs 1-3 establish notation and the concept of a variable quantity, which “encompasses within itself absolutely all numbers, both positive and negative, integers and rationals, irrationals and transcendentals” — and even zero and complex numbers (source: chapter1, §3).

Paragraph 4 gives the pivotal definition: a function of a variable is an analytic expression composed from the variable, numbers, and constants. See function.

Paragraphs 6-9 develop the classification-of-functions:

• algebraic vs. transcendental (§7),
• algebraic split into non-irrational vs. irrational, with irrational further split into explicit and implicit (§8),
• non-irrational split into polynomial and rational (§9).

Paragraphs 10-15 treat single-valued-and-multi-valued-functions, introducing $n$-valued functions defined by degree-$n$ polynomial equations with single-valued coefficients, together with the Vieta-style relations between the values and the coefficients.

Paragraphs 16-17 note the reciprocity of functional relations: if $y$ is a function of $z$, then $z$ is a function of $y$; and if $y$ and $x$ are both functions of $z$, each is a function of the other.

Paragraphs 18-25 develop even-and-odd-functions, including the multiplicative rules (even $\times$ odd = odd, odd $\times$ odd = even, etc.).

Paragraph 26 closes with similar-functions: $Y$ and $Z$ are similar functions of $y$ and $z$ when each is built from its variable by the same formal expression.

Notable points

• Euler admits complex values into the domain of a variable. The square root $\sqrt{9-z^2}$, though bounded by 3 on the reals, takes every value when $z$ is allowed to be complex, e.g. $z=5i$ (source: chapter1, §5).
• “Apparent functions” like $z^0$, $1^z$, and $\frac{a^2-az}{a-z}$ look like functions of $z$ but are actually constants (source: chapter1, §5).
• An expression with a transcendental constant but no transcendental operation on the variable is still algebraic: $c+z$, $c{z}^2$, and $4z^c$ (with $c$ the circumference of a unit-radius circle) count as algebraic functions of $z$ (source: chapter1, §7).
• A definition by an unsolvable polynomial equation, such as $Z^6=a{z}^2{Z}^3-b{z}^4{Z}^2+c{z}^{3}Z-1$, still counts as defining $Z$ as an (implicit) algebraic function of $z$ (source: chapter1, §7).
• If $n$ is odd, $P^{m/n}$ has a single real value and may be treated as single-valued; if $n$ is even and $m/n$ is in lowest terms, $P^{m/n}$ is two-valued (source: chapter1, §15).

Why this chapter matters

This is the chapter where the modern idea of “function” takes a recognizable shape. Euler’s definition as an analytic expression dominated 18th-century analysis and was only later displaced by the set-theoretic notion of a rule or mapping. The parity and multi-valuedness notions introduced here return throughout the Introductio.

Related pages

• function
• variable-and-constant
• classification-of-functions
• single-valued-and-multi-valued-functions
• even-and-odd-functions
• similar-functions