Definition:Proportion

Definition

Two real variables $x$ and $y$ are proportional iff one is a constant multiple of the other:

$\forall x, y \in \R: x \propto y \iff \exists k \in \R, k \ne 0: x = k y$

Inverse Proportion

Two real variables $x$ and $y$ are inversely proportional iff their product is a constant:

$\forall x, y \in \R: x \propto \dfrac 1 y \iff \exists k \in \R, k \ne 0: x y = k$

Joint Proportion

Two real variables $x$ and $y$ are jointly proportional to a third real variable $z$ iff the product of $x$ and $y$ is a constant multiple of $z$:

$\forall x, y \in \R: x y \propto z \iff \exists k \in \R, k \ne 0: x y = k z$

Constant of Proportionality

The constant $k$ is known as the constant of proportion, or (more common nowadays, but uglier) constant of proportionality.

Euclid's Definitions

As Euclid defined it:

Let magnitudes which have the same ratio be called proportional.

(The Elements: Book V: Definition $6$)

and:

Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.

(The Elements: Book VII: Definition $20$)

That is, if $a$ is to $b$ as $c$ is to $d$, that is:

$a : b = c : d$

where $a : b$ is the ratio of $a$ to $b$, then $a, b, c, d$ are proportional.

The definition is unsatisfactory, as the question arises: "proportional to what?"

Continuously Proportional

Four magnitudes $a, b, c, d$ are continuously proportional if $a : b = b : c = c : d$.

Sources

• Euclid: The Elements (c. 300 B.C.E.): Book $\text{V}, \ \text{VII}$
• Isaac Asimov: Understanding Physics (1966): $\text{I}$: Chapter $2$