sphere 471

exists a general formula akin to the quadratic formula

for quadratic equations that solves all fifth (or higher)

degree equations.

The work of É

VARISTE

G

ALOIS

(1811–32) provided

the first important steps toward understanding the gen-

eral properties of solutions to polynomial equations,

and Norwegian mathematician N

IELS

H

ENRIK

A

BEL

(1802–29) provided the first proof that there can be no

general formula for the solution of fifth-degree equations

via radicals. This is surprising given that the

FUNDAMEN

-

TAL THEOREM OF ALGEBRA

assures that every fifth-

degree equation has precisely five solutions. Abel’s result

shows then that, in general, not every solution can be

expressed in terms of a finite application of simple arith-

metic operations on the coefficients of the equation.

As mentioned earlier, some special quintics do have

solutions by radicals. For example, A

BRAHAM

D

E

M

OIVRE

(1667–1754) showed that the solutions of a

fifth-degree equation of the form:

(this polynomial is today known as De Moivre’s quin-

tic) has five solutions given by:

where is a fifth root of unity and kruns

through 0, 1, 2, 3, and 4.

See also G

IROLAMO

C

ARDANO

;

CUBE ROOT

/

NTH

ROOT

;

SQUARE ROOT

; N

ICCOLÒ

T

ARTAGLIA

.

Spearman’s method See

RANK CORRELATION

.

sphere The closed surface in three-dimensional space

consisting of all those points that are a fixed distance r

from a given point Ois called a sphere. The length ris

called the radius of the sphere, and the point Othe

center of the sphere. The

DISTANCE FORMULA

shows

that if Ohas C

ARTESIAN COORDINATES

O= (a,b,c),

then the equation of a sphere is given by:

(x– a)2+ (y– b)2+ (z– c)2= r2

It is often convenient to regard a sphere as centered

about the origin, in which case its equation reduces sim-

ply to x2+ y2+ z2= r2. This is the three-dimensional

analog to the equation of a

CIRCLE

of radius rin two-

dimensions: x2+ y2= r2. (The one-dimensional analog,

x2= r2, gives two points x= rand x= –ron the number

line, and the four-dimensional analog given by x2+ y2+

z2+ w2= r2is a hypersphere in the fourth

DIMENSION

.)

Any plane through the center of a sphere divides the

sphere into two hemispheres. The curve of intersection of

such a plane with the sphere is a circle on the surface of

the sphere called a great circle. (The equator of the

E

ARTH

, for instance, is a great circle.) Any other circle

drawn on the surface of the sphere is called a small circle.

Any line segment connecting two points on the sur-

face of the sphere and passing through the center of the

sphere is called a

DIAMETER

, and the two endpoints of

any diameter are called

ANTIPODAL POINTS

. The famous

Borsuk-Ulam theorem asserts the following:

If fand gare any two continuous functions on

the surface of a sphere (such as the air temper-

ature and air pressure at the surface of the

Earth), then there must exist a pair of antipo-

dal points Pand Qso that f(P) = f(Q) and g(P)

= g(Q).

The sphere can be regarded as a

SOLID OF REVOLU

-

TION

, obtained by rotating a circle about its diameter

one full revolution. As such, P

APPUS

’

S THEOREMS

show

that the

VOLUME

Vand the surface area Aof a sphere

of radius rare given by the formulae:

A= πr2

A= 4πr2

These formulae can also be obtained by the techniques

of

INTEGRAL CALCULUS

. (Calculus also shows that the

volume of a four-dimensional hypersphere of radius ris

V= π2r4.) One can also use calculus to establish the

following surprising result:

If a spherical loaf of bread is sliced into nslices

of equal thickness, then the surface area of

1

–

2

4

–

3