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 the concept of proportion, in order to specifically distinguish it from inverse proportion.
The terms:
can also be seen.
The term proportion is often known nowadays by the less elegant and more cumbersome word proportionality.
Also see
- Results about proportion can be found here.
Sources
- 1966: Isaac Asimov: Understanding Physics ... (previous) ... (next): $\text {I}$: Motion, Sound and Heat: Chapter $2$: Falling Bodies: Acceleration (Footnote $*$)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): proportional
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): variation: 1. (mutual variation)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): proportional
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): variation: 1. (mutual 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): proportion