Apollonius of Perga 15

regions on a unit

SPHERE

. A

SOLID ANGLE

of a region is

simply the measure of surface

AREA

of that region pro-

jected onto a unit sphere. Units of solid angle are called

steradians. The full surface area of the sphere is 4π

steradian.

See also B

ABYLONIAN MATHEMATICS

;

DIHEDRAL

;

SLOPE

;

TRIANGLE

;

TRISECTING AN ANGLE

.

angle trisection See

TRISECTING AN ANGLE

.

annulus (plural, annuli) An annulus is the region

between two

CONCENTRIC

circles in a plane. The

AREA

of the annulus is the difference of the areas of

the two circles.

If a cyclist rides a perfect circle, the region between

the tracks made by the front and rear wheels is an

annulus. If the bicycle is rfeet long (axel to axel), then,

surprisingly, the area of this annulus is πr2feet squared,

irrespective of the size of the circle the cyclist traces.

antidifferentiation (integration) The process of

finding a function with a given function as its

DERIVA

-

TIVE

is called antidifferentiation. For example, x2is an

antiderivative of 2x, since . The

MEAN

VALUE THEOREM

shows that two antiderivatives of

the same function differ only by a constant. Thus all

the antiderivatives of 2x, for example, are functions

of the form x2+ C.

The antiderivative of a function f(x) is denoted

∫f(x)dx and is called the indefinite integral of f. It is

defined up to a constant, and so we write, for example,

∫2x dx = x2+ C. (The constant Cis referred to as a

“constant of integration.”) The notation is deliberately

suggestive of a definite integral of a function, ∫b

af(x)dx,

for the area under the curve y= f(x) over the

INTERVAL

[a,b]. The

FUNDAMENTAL THEOREM OF CALCULUS

shows that the two notions are intimately connected.

See also

INTEGRAL CALCULUS

.

antilogarithm See

LOGARITHM

.

antipodal points (antipodes) Two points on a

SPHERE

at the opposite ends of a diameter are said to be

antipodal. For example, the north and south poles are

antipodal points on the E

ARTH

, as are any two points

EQUIDISTANT

from the equator, with longitudes differ-

ing by 180°.

The famous Borsuk-Ulam theorem, first conjec-

tured by Stanislaw Ulam and then proved by Karol

Borsuk in 1933, states:

Let fbe a continuous function that assigns two

numerical values to each and every point on

the surface of a sphere. Then there must exist

two antipodal points which are assigned pre-

cisely the same pair of values.

An amusing interpretation reads:

At any instant there exist two antipodal points

on the Earth’s surface of precisely the same air

temperature and air pressure.

Although the proof of this theorem is difficult, a one-

dimensional version of the result follows as an easy

consequence of the

INTERMEDIATE

-

VALUE THEOREM

.

apex (plural, apices) The point at the top of a

POLY

-

GON

or a

POLYHEDRON

, such as the vertex of a triangle

opposite its

BASE

or the vertex of a pyramid, is called

the apex of the figure. The distance from the base of

the figure to its apex is called the height of the figure.

Apollonius of Perga (ca. 262–190

B

.

C

.

E

.) Greek Geo-

metry Born in Perga, Greek Ionia, now Antalya,

Turkey, Apollonius worked during the Golden Age of

Greek mathematics and has been referred to throughout

history as the Great Geometer. His famous work, The

Conics, written in eight volumes, greatly influenced the

development of mathematics. (The names

ELLIPSE

,

PARABOLA

, and

HYPERBOLA

for the three

CONIC SEC

-

TIONS

, for instance, are said to have been coined by

Apollonius.) Copies of the first four volumes of this

work, written in the original Greek, survive today. Ara-

bic translations of the first seven volumes also exist.

Little is known of Apollonius’s life other than what

can be gleaned from incidental comments made in the

prefaces of his books. As a young man it is known that

he traveled to Alexandria to study with the followers of

E

UCLID

, who then introduced him to the topic of conics.