Variable and Constant
Summary: Euler’s two primitive kinds of quantity. A constant keeps one value; a variable is a “genus” that ranges over all numbers of all types, including complex numbers.
Sources: chapter1
Last updated: 2026-04-23
Constant quantity
A constant quantity is a determined quantity which always keeps the same value. (source: chapter1, §1)
Constants are represented by the initial letters of the alphabet: , , , etc. In classical algebra these early letters denote known quantities; in analysis Euler uses them for fixed (but possibly arbitrary) parameters.
Variable quantity
A variable quantity is one which is not determined or is universal, which can take on any value. (source: chapter1, §2)
Variables are denoted by the final letters of the alphabet: , , , etc. Euler describes a variable as “a genus in which are contained all determined quantities” — it is a universal placeholder, not a specific number.
A variable can be determined in infinitely many ways, since any number may be substituted for it. The scope of “any number” is striking:
A variable quantity encompasses within itself absolutely all numbers, both positive and negative, integers and rationals, irrationals and transcendentals. Even zero and complex numbers are not excluded from the signification of a variable quantity. (source: chapter1, §3)
This liberal scope is what allows Euler, later in §5, to argue that attains every value once complex inputs are permitted.
Notational convention
| Role | Letters used |
|---|---|
| Constants | (start of alphabet) |
| Variables | (end of alphabet) |
Euler also uses Greek letters as a second family of constants, e.g. for denominator coefficients in a rational function (source: chapter1, §9).
Why this matters
The variable/constant distinction is what makes the notion of a function well-defined: a function of is an expression in which everything non- is a constant (source: chapter1, §4).