Proportion

The mathematical and geometrical rules conditioning the relationship of the parts to each other and to the whole.

Since the days of the Ancient Egyptian civilization, artists have employed some system of proportional canon. The Greek canon was established with the famous statue of Doryphorus by POLYCLITUS of SAMOS (5th century BC), and was modified by LYSIPPUS (fl. c.360-316 BC).

Further modifications were introduced by: the Roman VITRUVIUS (fl.46-30 BC), the medieval mason VILLARD de HONNECOURT (fl. 1225-1235 AD), Italian architect and theorist LEON BATTISTA ALBERTI (1404-1472), LEONARDO DA VINCI (1452-1519) and ALBRECHT DüRER (1471-1528), each reflecting particular aesthetic aims.

In mathematics, two varying quantities are said to be in a relation of proportionality, multiplicatively connected to a constant; that is, when either their ratio or their product yields a constant. The value of this constant is called the coefficient of proportionality or proportionality constant.

• If the ratio (y/x) of two variables (x and y) is equal to a constant (k = y/x), then the variable in the numerator of the ratio (y) can be product of the other variable and the constant (y = k ⋅ x). In this case y is said to be directly proportional to x with proportionality constant k. Equivalently one may write x = 1/k ⋅ y; that is, x is directly proportional to y with proportionality constant 1/k (= x/y). If the term proportional is connected to two variables without further qualification, generally direct proportionality can be assumed.
• If the product of two variables (x ⋅ y) is equal to a constant (k = x ⋅ y), then the two are said to be inversely proportional to each other with the proportionality constant k. Equivalently, both variables are directly proportional to the reciprocal of the respective other with proportionality constant k (x = k ⋅ 1/y and y = k ⋅ 1/x).

If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., a/b = x/y = ⋯ = k (for details see Ratio).

Inverse proportionality

The concept of inverse proportionality can be contrasted with direct proportionality. Consider two variables said to be “inversely proportional” to each other. If all other variables are held constant, the magnitude or absolute value of one inversely proportional variable decreases if the other variable increases, while their product (the constant of proportionality k) is always the same. As an example, the time taken for a journey is inversely proportional to the speed of travel.

Formally, two variables are inversely proportional (also called varying inversely, in inverse variation, in inverse proportion, in reciprocal proportion)^{[citation needed]} if each of the variables is directly proportional to the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant.^[2] It follows that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that

{\displaystyle y={\frac {k}{x}},}

or equivalently, {\displaystyle xy=k.} Hence the constant “k” is the product of x and y.

The graph of two variables varying inversely on the Cartesian coordinate plane is a rectangular hyperbola. The product of the x and y values of each point on the curve equals the constant of proportionality (k). Since neither x nor y can equal zero (because k is non-zero), the graph never crosses either axis.

Hyperbolic coordinates

The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particular ray and the constant of inverse proportionality that specifies a point as being on a particular hyperbola.