Definition:Proportion

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Definition

Two real variables $x$ and $y$ are proportional if and only if 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 if and only if 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 $z$ if and only if 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 Proportion

The constant $k$ is known as the constant of proportion.


Euclid's Definitions

In the words of Euclid:

Let magnitudes which have the same ratio be called proportional.

(The Elements: Book $\text{V}$: 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 $\text{VII}$: 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?"


Perturbed Proportion

Let $a, b, c$ and $A, B, C$ be magnitudes.

$a, b, c$ are in perturbed proportion to $A, B, C$ if and only if:

$a : b = B : C$
$b : c = A : B$


In the words of Euclid:

A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.

(The Elements: Book $\text{V}$: Definition $18$)


Continued Proportion

Four magnitudes $a, b, c, d$ are in continued proportion if and only if $a : b = b : c = c : d$.


Also known as

The term direct proportion can frequently be seen for this concept, in order to specifically distinguish it from inverse proportion.

The term direct variation can also be seen.


The term proportion is more usually known nowadays by the less elegant and more cumbersome word proportionality.


Sources