Classification of Functions

Summary: Euler organizes functions by which operations are used to build them and how those operations involve the variable. The result is a tree: algebraic vs. transcendental, then non-irrational vs. irrational, then polynomial vs. rational.

Sources: chapter1

Last updated: 2026-04-23

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The tree

function
|-- algebraic           (only algebraic operations on the variable)
|   |-- non-irrational  (no radicals involving the variable)
|   |   |-- polynomial  (non-negative integer exponents of $z$ only)
|   |   `-- rational    ($z$ appears in denominators or with negative exponents)
|   `-- irrational      (variable affected by radical signs)
|       |-- explicit    (radicals written out)
|       `-- implicit    (defined by an unsolved polynomial equation)
`-- transcendental      (transcendental operation actually affects the variable)

The splits are defined by:

• Algebraic vs. transcendental (§7): a function is algebraic if it uses only the algebraic operations listed in function — addition, subtraction, multiplication, division, integer powers, roots, solving polynomial equations. It is transcendental if a transcendental operation (exponential, logarithm, etc.) actually affects the variable. An expression with a transcendental constant but no transcendental operation on the variable is still algebraic; Euler’s examples are $c+z$, $c{z}^2$, $4z^c$ with $c$ the circumference of a unit-radius circle (source: chapter1, §7).
• Non-irrational vs. irrational (§8): non-irrational uses only addition, subtraction, multiplication, division, and integer-exponent powers of $z$; irrational has the variable inside a radical.
• Polynomial vs. rational (§9): polynomial has no negative exponents and no $z$ in denominators; rational allows either.

General forms

Polynomial:

$$a+bz+c{z}^2+d{z}^3+e{z}^4+f{z}^5+\dots$$

Every polynomial function fits this template (source: chapter1, §9).

Rational (numerator and denominator are polynomials):

$$\frac{a+bz+c{z}^2+d{z}^3+\dots }{\alpha +\beta z+\gamma z^2+\delta z^3+\dots }$$

Every rational function reduces to this form by combining fractions over a common denominator. The coefficients $a,b,c,\dots$ and $\alpha ,\beta ,\gamma ,\dots$ may be “positive or negative, integers or fractions, rational or irrational, or even transcendental” without changing the classification (source: chapter1, §9).

Irrational functions

Irrational functions are split into:

• Explicit: written with radical signs, e.g. $\sqrt{z}$, $a+\sqrt{a^2-z^2}$, $(a-2z+z^2{)}^{1/3}$, $\frac{a^2-z\sqrt{a^2+z^2}}{a+z}$ (source: chapter1, §8). These arise from the extraction of roots operation.
• Implicit: defined by an equation that cannot be solved explicitly by radicals, e.g. $Z$ defined by $Z^7=az$ or by the irreducible equation $Z^6=a{z}^2{Z}^3-b{z}^4{Z}^2+c{z}^{3}Z-1$ (source: chapter1, §7, §8). These arise from the solution of equations operation.

Euler is careful to distinguish solvability in principle from solvability in current practice: “common algebra has not yet developed to such a degree of perfection” (source: chapter1, §8). This is an 18th-century remark, written before Ruffini (1799) and Abel (1824) proved the impossibility of solving the general quintic by radicals.

Edge case: exponents

An expression like $z^c$, where $c$ is the transcendental constant $\pi$ (unit-circle circumference = $2\pi$, but Euler here uses $c$ for the circumference itself), is strictly algebraic under Euler’s rule because the transcendental enters only as a constant. Some authors prefer to call such expressions “intercendental” rather than algebraic; Euler himself notes the dispute but treats it as unimportant (source: chapter1, §7).

Extension to several variables (Chapter 5, §79–§80)

The same tree is used verbatim for multivariate functions. A function of $y,z,\dots$ is transcendental if a transcendental operation affects some variable; a “mixed” case is allowed where the transcendental involves only some variables — e.g. $z^2+y\log z$ is transcendental in $y,z$ jointly but becomes algebraic in $y$ when $z$ is fixed (source: chapter5, §79). Algebraic multivariate functions split the same way: irrational (explicit or implicit) vs. non-irrational, and non-irrational into polynomial and rational. The general polynomial in $y,z$ is

$\alpha +\beta y+\gamma z+\delta y^2+ϵyz+\zeta z^2+\eta y^3+\theta y^{2}z+\iota y{z}^2+\kappa z^3+\cdots$

and the general rational function is $P/Q$ for polynomials $P,Q$ (source: chapter5, §80). Multivariate polynomials and rational functions admit the additional classifications of homogeneity, order, and reducibility developed in chapter-5-on-functions-of-two-or-more-variables — see homogeneous-function and reducible-polynomial.

Related pages

• function
• variable-and-constant
• single-valued-and-multi-valued-functions
• chapter-1-on-functions-in-general
• functions-of-several-variables
• homogeneous-function
• reducible-polynomial