Oughtred, William 369

orientation In mathematics, the term orientation is

used to refer to a sense of direction or rotation. For

example, a line that is labeled with a direction is said to

be oriented, and a closed loop drawn in the plane can

be assigned either a clockwise or counterclockwise ori-

entation. In two- and three-dimensional space, coordi-

nate axes are said to be either positively or negatively

oriented (and are called

RIGHT

-

HANDED

/

LEFT

-

HANDED

SYSTEM

s), and a surface, such as a

SPHERE

for example,

is said to be orientable, since at every location on the

surface there is, loosely speaking, a well-defined notion

of “up.” (For example, as inhabitants of the surface of

the Earth we define “up” to the direction pointing

away from the center of the Earth.) Not all surfaces are

orientable. The M

ÖBIUS BAND

, for example, is a surface

that cannot be oriented.

orthogonal (perpendicular) This term is used in any

setting to describe two geometric constructs that meet

at right angles. For example, two curves, or straight

lines, are orthogonal if they intersect at right angles,

that is, the angle between the two

TANGENT

s to the

curves at the point of intersection is 90°. Two

VECTOR

s

are orthogonal if the angle between them is 90°(and

consequently their

DOT PRODUCT

is zero). Two surfaces

can also be said to be orthogonal. For example, a plane

passing through the center of a sphere intersects the

surface of the sphere orthogonally—the tangent plane

to the sphere at any point of intersection is perpendicu-

lar to the given plane.

The term orthogonal is also used in some general-

ized settings. For example, two functions fand gare

said to be orthogonal over the interval [a,b] if the inte-

gral ∫b

af(x)g(x)dx is zero. (The dot product a · b of two

vectors a= < a1, a2,…, an> and b= < b1, b2,…, bn> is

the sum of the vector components multiplied together:

a· b= a1b1+ a2b2+…+ anbn. The above integral is a

generalized sum of the components of the functions

multiplied together.) The functions sin(x) and cos(x),

for example, are orthogonal over the interval [0, 2π].

This is an important observation for the development

of F

OURIER SERIES

.

A

MATRIX

is said to be orthogonal if its rows repre-

sent vectors that, taken any two at a time, have dot

product equal to zero.

See also

NORMAL TO A CURVE

;

NORMAL TO A

PLANE

;

NORMAL TO A SURFACE

.

Osborne’s rule Mathematicians have observed that

each trigonometric identity yields an identity for

HYPERBOLIC FUNCTIONS

if we simply:

1. Replace each trigonometric function with its hyper-

bolic analog.

2. Change the sign of any term involving the product

of two hyperbolic sines (sinh).

This principle is called Osborne’s rule. For example,

from the trigonometric identity cos(x+ y) = cosxcosy

– sinxsiny, we obtain the hyperbolic identity

cosh(x+y) = cosh xcoshy+ sinhxsinhy. From 1 +

tan2x= sec2x, which is , follows

, or 1 – tanh2x= sech2x. This

principle works because, by E

ULER

’

S FORMULA

and

the definition of the hyperbolic functions, cosx=

= cosh(ix) and sin x= = –isinh(ix).

Thus any identity that holds for sines and cosines will

also hold for cosh and sinh, except a factor of i2= –1

will alter the sign of a product of two hyperbolic

sines.

Oughtred, William (1574–1660) British Logarithms

Born March 5, 1574, English mathematician William

Oughtred is best remembered for his work in develop-

ing and designing the calculating device known as a

SLIDE RULE

.

At the turn of the 17th century, scientists were

excited by the recent discovery of

LOGARITHM

s as an

aid for converting tedious computations of multiplica-

tion and division into simpler operations of addition

and subtraction. In 1620 English mathematician

Edmund Gunter plotted a

LOGARITHMIC SCALE

along a

2-ft-long ruler and showed how a pair of calipers could

be used to physically add and subtract lengths, and

therefore provide, for the first time, a purely mechanical

means of computing products and quotients. Inspired

by the work of Gunter, Oughtred devised a simpler

device consisting of two sliding rulers that accomplished

the same feat. In 1632 he published Circles of Propor-

tion and the Horizontal Instrument, a short book

describing slide rules (and sundials). Oughtred’s slide

rule became the modern-day equivalent of today’s