Functions of Several Variables

Summary: Starting in §77, Euler introduces several independent variables and defines a function of them as any expression built from them. The algebraic/transcendental and single/multi-valued classifications from Chapter 1 carry over verbatim, and §82 gives his first statement of the equation-counting rule behind implicit functions.

Sources: chapter5

Last updated: 2026-04-23

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Independent variables

Through Chapter 4 every “variable” had been a function of one independent $z$. In §77 Euler changes the setting: quantities $x,y,z,\dots$ vary independently. Fixing a value of one leaves the others “completely unrestricted” (source: chapter5, §77). A function of several variables is any expression composed from them — e.g. $x^3+xyz+a{z}^3$ is a function of $x,y,z$; it becomes a function of $x,y$ when $z$ is fixed, and a function of $x$ alone when $y$ and $z$ are both fixed (source: chapter5, §78).

The admissible values form what Euler calls an “infinity of infinite determinations” — a two-variable function admits infinitely many choices for each variable, so “infinitely many” per value of the other. A three-variable function has one more level of infinity, and so on.

Classification carries over

• Algebraic vs. transcendental (§79): same split as in classification-of-functions. A transcendental operation in a multivariate function may involve all, some, or only one of the variables — e.g. $z^2+y\log z$ is transcendental in $y,z$ jointly, but algebraic in $y$ once $z$ is fixed (source: chapter5, §79).
• Irrational vs. non-irrational; polynomial vs. rational (§80): a non-irrational function is free of radicals; if no variable appears in a denominator it is a polynomial, otherwise a rational function $P/Q$. The general polynomial in $y,z$ is $\alpha +\beta y+\gamma z+\delta y^2+ϵyz+\zeta z^2+\cdots$ (source: chapter5, §80).
• Explicit vs. implicit irrational (§80): an implicit irrational function is given by an unsolvable polynomial equation, e.g. $V^5=(ayz+z^3)V^2+(y^4+z^4)V+y^5+2ay{z}^3+z^5$.
• Single- vs. multi-valued (§81): as in single-valued-and-multi-valued-functions, $V$ is $k$-valued if $V^k-P{V}^{k-1}+Q{V}^{k-2}-\cdots =0$ for single-valued $P,Q,\dots$ Non-irrational functions are automatically single-valued (source: chapter5, §81).

§82 — Implicit definition by equations

Setting a multivariate function equal to zero (or to a constant, or to another function) ties its variables together: each becomes a function of the rest. The general rule Euler states (source: chapter5, §82):

The number of equations determines the number of functions defined.

Concretely:

• One equation in two variables $y,z$ makes $y$ a function of $z$ (and vice versa).
• One equation in three variables $x,y,z$ makes any one of them a function of the other two.
• Two equations in three variables make a pair of them functions of the third.
• In general, $k$ equations in $n$ variables leave $n-k$ degrees of freedom.

This is a very early — and still informal — statement of the implicit-function-and-equation-counting idea. Euler does not ask when the resulting function is single-valued, or when the implicit relation is solvable at all.

Related pages

• classification-of-functions
• single-valued-and-multi-valued-functions
• implicit-function-and-equation-counting
• homogeneous-function
• chapter-5-on-functions-of-two-or-more-variables