# Definition:Proportion

## 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

- 1966: Isaac Asimov:
*Understanding Physics*... (previous) ... (next): $\text {I}$: Motion, Sound and Heat: Chapter $2$: Falling Bodies: Acceleration (Footnote $*$) - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**direct variation** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**direct** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**direct variation** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**varies directly, varies inversely**