Chapter 5: On Functions of Two or More Variables

Summary: Euler generalizes the single-variable framework of Chapters 1–4 to functions of several independent variables, carries the algebraic/transcendental classification over verbatim, and introduces the central new concept — homogeneous functions — together with the substitution $y=uz$ that reduces every bivariate homogeneous function of degree $n$ to $z^n\cdot f(u)$.

Sources: chapter5

Last updated: 2026-04-23

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Overview

Through Chapter 4 every “variable” had implicitly been a function of a single independent $z$. Chapter 5 (§77) declares the opposite setting: several quantities $x,y,z$ may vary independently, and an expression built from them is a function of several variables (source: chapter5, §77–§78). A function of $k$ variables admits an “infinity of infinite determinations” — one chooses one variable and still has a full $(k-1)$-dimensional freedom (source: chapter5, §78). See functions-of-several-variables.

The old classification — algebraic vs. transcendental, with algebraic split into polynomial, rational, and irrational — transfers without change (§79–§80). Multi-valuedness is still defined by single-valued coefficients in a polynomial equation $V^k-P{V}^{k-1}+Q{V}^{k-2}-\cdots =0$ (§81). Setting a multivariate function equal to zero (or a constant, or another function) implicitly defines each variable as a function of the rest — Euler’s first statement of the implicit-function-and-equation-counting idea and the “count equations, count dependent variables” heuristic (§82).

The substantive new material is the theory of homogeneous-functions (§83–§91), together with its counterparts for heterogeneous-functions (§92–§93) and the classification of polynomials by order and reducibility (§94–§95). The key theorem, §88, shows that every homogeneous function of degree $n$ in two variables is $z^{n}f(y/z)$ — the theoretical foundation for the homogeneous-substitution trick Euler used empirically in Chapter 3.

See also: functions-of-several-variables, implicit-function-and-equation-counting, homogeneous-function, heterogeneous-function, order-of-a-polynomial, reducible-polynomial.

Structure of the chapter

§77–§78 — Independent variables and functions of several variables

Up to now every “variable” was secretly a function of one $z$ (source: chapter5, §77). Now $x,y,z,\dots$ may take values independently: fixing $z$ leaves the rest “completely unrestricted.” A function of several variables is any expression composed from them — e.g. $x^3+xyz+a{z}^3$ is a function of $x,y,z$; fix $z$ and it becomes a function of $x,y$; fix $z$ and $y$ and it becomes a function of $x$ (source: chapter5, §78). The admissible values of an $n$-variable function form an “infinity of infinite determinations” of multiplicity $n$.

§79–§80 — Classification carries over

The algebraic/transcendental split is the same as in Chapter 1 (see 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 becomes algebraic in $y$ once $z$ is fixed (source: chapter5, §79). Algebraic functions split into irrational and non-irrational; the latter into polynomials (no variable in a denominator) and rationals $P/Q$ (ratio of polynomials) (source: chapter5, §80). Irrational functions may be explicit (a radical sign) or implicit (given by an unsolvable polynomial equation), e.g. $V^5=(ayz+z^3)V^2+(y^4+z^4)V+y^5+2ay{z}^3+z^5$.

§81 — Multi-valued functions

Multi-valuedness is defined exactly as in Chapter 1 (see single-valued-and-multi-valued-functions): $V$ is $k$-valued if $V^k-P{V}^{k-1}+Q{V}^{k-2}-\cdots =0$ with single-valued $P,Q,\dots$ (source: chapter5, §81). Non-irrational functions are automatically single-valued.

§82 — Implicit definition from equations

Setting a function of $y$ and $z$ equal to zero makes each of $y,z$ a function of the other — previously independent, now tied. The same works if the function equals a constant, or equals another function. Extending: one equation in three variables makes any one of them a function of the remaining two; two equations define a pair of variables as functions of the rest; in general the number of equations determines the number of functions defined (source: chapter5, §82). This is Euler’s first statement of the implicit-function-and-equation-counting principle.

§83 — Homogeneous vs. heterogeneous

A function is homogeneous if every term has the same total degree, heterogeneous if different degrees appear. Each variable counts for one degree; constants count for zero; products of $k$ variables (repeats allowed) count for degree $k$. So $\alpha y,\beta z$ have degree 1; $\alpha y^2,\beta yz,\gamma z^2$ have degree 2; and so on (source: chapter5, §83). See homogeneous-function.

§84–§85 — Polynomial and rational cases

Homogeneous polynomials of given degree have the obvious form: $\alpha y+\beta z$ (degree 1); $\alpha y^2+\beta yz+\gamma z^2$ (degree 2); etc. A constant function counts as degree zero (source: chapter5, §84).

A rational function $P/Q$ is homogeneous if both $P$ and $Q$ are, with degree $\mathrm{deg}P-\mathrm{deg}Q$. Negative and zero degrees are admitted: $y/z^2$ has degree $-1$, $(y+z)/(y^4+z^4)$ has degree $-3$, $1/(y^5+ay{z}^4)$ has degree $-5$, and $(y^3+z^3)/(y^{2}z)$ has degree 0 (source: chapter5, §85).

§86 — Irrational case

The degree extends to irrational functions by the rule: if $P$ is homogeneous of degree $n$, then $P^{\mu /\nu }$ is homogeneous of degree $(\mu /\nu )n$ (source: chapter5, §86). So $\sqrt{y^2+z^2}$ has degree 1; $(y^9+z^9{)}^{1/3}$ has degree 3; $(yz+z^2{)}^{1/2}$ has degree 1; ${\left((y^2+z^2)/(y^4+z^4)\right)}^{1/1}$ has degree $-2$; and so forth. Sums and ratios of homogeneous pieces of matching degree are again homogeneous of that degree.

§87 — Implicit irrational case

If $V$ satisfies $V^k+P{V}^{k-1}+Q{V}^{k-2}+\cdots +R=0$ with $P,Q,\dots ,R$ polynomials in $y,z$, then $V$ is homogeneous (of some degree $n$) iff $P$ has degree $n$, $Q$ has degree $2n$, $R$ has degree $kn$, etc. — each coefficient matches the degree required for the whole equation to be homogeneous (source: chapter5, §87). Example: $V^5+(y^4+z^4)V^3+a{y}^{8}V-z^{10}=0$ has $V$ homogeneous of degree 2.

§88 — Euler’s reduction $y=uz$

Theorem. If $V$ is homogeneous of degree $n$ in $y,z$, then under $y=uz$ one has

$V(y,z)=z^n\cdot f(u),u=y/z.$

Proof sketch. Every term of $V$ has total degree $n$ in $y,z$; replacing $y$ by $uz$ converts joint degree into degree in $z$ alone, so each term carries a factor $z^n$ (source: chapter5, §88). Euler checks all three cases:

• Polynomial: $V=\alpha y^3+\beta y^{2}z+\gamma y{z}^2+\delta z^3\Rightarrow V=z^3(\alpha u^3+\beta u^2+\gamma u+\delta )$.
• Rational: $V=(\alpha y+\beta z)/(y^2+z^2)\Rightarrow V=z^{-1}(\alpha u+\beta )/(u^2+1)$.
• Irrational: $V=(y+\sqrt{y^2+z^2})/(z\sqrt{y^3+z^3})\Rightarrow V=z^{-3/2}(u+\sqrt{u^2+1})/\sqrt{u^3+1}$.

This theorem is the theoretical backbone of the homogeneous-substitution trick from Chapter 3, §52–§58.

§89 — Degree zero: a function of one variable

When $n=0$, the factor $z^0=1$ drops out: a homogeneous function of degree zero in $y,z$ is a function of $u=y/z$ alone (source: chapter5, §89). Example: $(y+z)/(y-z)=(u+1)/(u-1)$; $(y-\sqrt{y^2-z^2})/z=u-\sqrt{u^2-1}$.

§90–§91 — Homogeneous bivariate polynomials factor linearly

A homogeneous polynomial of degree $n$ in $y,z$ factors as the product of $n$ linear pieces $\alpha y+\beta z$ (real or complex) (source: chapter5, §90–§91). Proof. After $y=uz$, the polynomial becomes $z^n$ times a polynomial in $u$ alone, which factors into linear pieces $\alpha u+\beta$; multiplying each by $z$ gives $\alpha y+\beta z$.

Thus every homogeneous polynomial of degree $n$ in two variables is reducible, a product of the right number of linear factors.

This property fails in three or more variables. The general homogeneous degree-2 form in three variables, $a{y}^2+byz+c{z}^2+dyx+ezx+f{x}^2$, does not generally factor as $(\alpha y+\beta z+\gamma x)(\delta y+ϵz+\zeta x)$ (source: chapter5, §91).

§92 — Heterogeneous functions; bifid, trifid, …

Heterogeneous functions are classified by how many distinct degrees occur among their terms. A bifid function has two: e.g. $y^5+2y^3{z}^2+y^2+z^2$ splits as (degree-5 part) + (degree-2 part). A trifid has three: e.g. $y^6+y^2{z}^2+z^4+y-z$ has parts of degrees 6, 4, 1. Some rational or irrational functions cannot be cleanly split this way at all — e.g. $(y^3+ayz)/(by+z^2)$ (source: chapter5, §92).

§93 — Reducing heterogeneous to homogeneous by substitution

Sometimes a substitution makes a heterogeneous function homogeneous. Examples (source: chapter5, §93):

• $y^5+z^{2}y+y^{3}z+z^3/y$ under $z=x^2$ becomes $y^5+x^{4}y+y^3{x}^2+x^6/y$ — homogeneous of degree 5 in $x,y$.
• $y+y^{2}x+y^3{x}^2+y^5{x}^4+a/x$ under $z=1/x$ becomes $y+y^2/z+y^3/z^2+y^5/z^4+az$ — homogeneous of degree 1.

No general criterion is given; Euler is content with examples.

§94 — Order of a polynomial

The order of a polynomial is the greatest degree of any single term. So $z^2+y^2+z^2+ay-a^2$ is of order 2 (even though the constant term has degree 0), and $y^4+y{z}^3-a{y}^{2}x+abyz-a^2{y}^2+b^4$ is of order 4 (source: chapter5, §94). Order is the classification relevant to the study of algebraic curves.

§95 — Reducible and irreducible polynomials

A polynomial is reducible if it is a product of two or more non-irrational factors, irreducible otherwise. Example: $y^4-z^4+2a{z}^3-2by{z}^2-a^2{z}^2+2abzy-b^2{y}^2=(y^2+z^2-az+by)(y^2-z^2+az-by)$. By §91, every homogeneous bivariate polynomial is reducible. In contrast, $y^2+z^2-a^2$ is irreducible. Reducibility is decided by examining divisors (source: chapter5, §95). See reducible-polynomial.

Notable points

• §77 marks a conceptual shift. The single-variable setting of Chapters 1–4 was implicitly a curve: one parameter, one-dimensional image. Several variables means several degrees of freedom — surfaces and higher — and the rest of Book I and Book II will develop both the algebraic theory (here and in later chapters) and the geometric applications to curves and surfaces.
• §82 is the first place in the Introductio where Euler states, in passing, the equation-counting rule: $k$ equations in $n$ variables leave $n-k$ free. The notion of “implicit function” is still informal — there is no discussion of when the resulting map is well defined or single-valued.
• §88 is the payoff. The reason the homogeneous-substitution $y=xz$ in §52–§58 worked is that it produced $z^{m-n}=$ rational$(x)$ — exactly what §88 guarantees. Euler now has the theorem under his belt: in Chapter 3 he used it as a procedure; here he names it.
• §91’s observation that bivariate homogeneous polynomials factor completely into linear factors, but trivariate ones do not, is an early piece of geometric intuition: a homogeneous polynomial in two variables cuts out a finite set of lines through the origin, while a homogeneous polynomial in three variables cuts out a projective curve, which need not split.
• §92–§93 are comparatively unsystematic — Euler records the classes without developing them further. The distinction will return when he studies algebraic curves by their equations of given order.

Why this chapter matters

Chapter 5 is short but structurally pivotal. It sets up the multivariate framework every later chapter on curves and surfaces will assume, and it supplies the missing theorem — the $y=uz$ reduction for homogeneous functions — that retroactively justifies the parametrization tricks of Chapter 3. The classifications introduced here (homogeneous/heterogeneous, order, reducibility) become the working vocabulary for the algebraic theory of curves that occupies much of the remainder of Book I.

Related pages

• functions-of-several-variables
• implicit-function-and-equation-counting
• homogeneous-function
• heterogeneous-function
• order-of-a-polynomial
• reducible-polynomial
• homogeneous-substitution
• classification-of-functions
• single-valued-and-multi-valued-functions
• chapter-3-on-the-transformation-of-functions-by-substitution