Implicit Function and Equation Counting
Summary: In Chapter 5, Euler observes that setting a relation among several variables makes each variable a function of the others, and that each independent equation reduces the number of free variables by one. This is his early formulation of the implicit-function viewpoint and the equation-counting heuristic.
Sources: chapter5
Last updated: 2026-04-24
The basic idea
If a function of several variables is set equal to zero, a constant, or another function, the variables are no longer independent: each is determined by the others (source: chapter5, §82).
Euler’s guiding rule is that the number of independent equations measures how many variables become dependent. With one equation in variables, one variable may be regarded as a function of the remaining ; with independent equations, variables become dependent on the others (source: chapter5, §82).
Why it matters
This is not yet a modern implicit-function theorem with hypotheses and proof. It is a structural heuristic for keeping track of freedom: each new relation reduces the dimension of the family of possible values by one (source: chapter5, §82).
In the context of the Introductio, this remark extends the Chapter 1 idea of a function beyond explicit formulas. A quantity can be a function of others even when it is given only through an equation rather than solved for directly.