474 spherical triangle

Pythagoras’s theorem

a

–

θ

Riemann and Klein proved that all but the fifth of

E

UCLID

’

S POSTULATES

hold in this model and, moreover,

that this model is consistent (that is, free of

CONTRA

-

DICTION

s). This establishes that the parallel postulate

cannot be logically deduced as a consequence of the

remaining axioms proposed by Euclid.

In spherical geometry all angles in triangles sum to

more than 180°, and the ratio of the circumference of

any circle to its diameter is greater than π(and this

value varies from circle to circle).

See also E

UCLIDEAN GEOMETRY

;

HYPERBOLIC GEOM

-

ETRY

; P

LAYFAIR

’

S AXIOM

.

spherical triangle A three-sided figure on the surface

of a

SPHERE

bounded by the arcs of three great circles is

called a spherical triangle. Unlike plane triangles, the

angles in a spherical triangle do not sum to the con-

stant value of 180°. In fact, the sum of angles is always

greater than 180°and can be as great as 540°. (This

occurs when all three sides of the spherical triangle lie

on the same great circle.)

A right spherical triangle has at least one right angle.

A birectangular triangle contains two right angles, and a

trirectangular triangle has three right angles. (For

instance, any triangle that connects the North Pole to

two points on the equator of the Earth is birectangular.

Such a triangle can also be trirectangular if the points on

the equator are positioned appropriately.) A spherical

triangle with no right angles is called oblique.

If the angles of a spherical triangle are A, B, and C,

measured in degrees, then the surface area of the trian-

gle is given by the formula:

Here Ris the radius of the sphere. The quantity A+ B

+ C– 180, usually denoted E, is called the spherical

excess of the triangle. If the three side-lengths of the tri-

angle are given by a, b, and c, then the spherical excess

can also be computed via the following remarkable for-

mula discovered by Swiss mathematician Simon

Lhuilier (1750–1840):

spiral of Archimedes (Archimedean spiral) Studied

by the great A

RCHIMEDES OF

S

YRACUSE

(ca. 287–212

B

.

C

.

E

.) in his work On Spirals, an Archimedean spiral is

the curve traced out by a point rotating about a fixed

point at a constant angular speed while moving away

from that point at a constant speed. (For example, it is

the path traced by a fly walking radially outward from

the center of a uniform rotating disc.) The equation of

such a spiral has a simple form in

POLAR COORDINATES

:

r= aθ

where ais a positive constant.

A curve given by an equation of the form r= is

called a hyperbolic spiral. These curves were studied by

Johann Bernoulli of the famous B

ERNOULLI FAMILY

.

Curves of the form appear as tightly wrapped

spirals. A curve given by logr= aθis called a logarith-

mic spiral.

See also

GOLDEN RATIO

.

square In geometry a square is a planar figure with

four equal straight sides and four interior right angles.

It is simultaneously a

RECTANGLE

, a

PARALLELOGRAM

,

and a rhombus.

A square with side-length xhas

AREA

x×x= x2,

which explains the term squaring for the operation of

raising a number or a variable to the second power.

Squares are solutions to

ISOPERIMETRIC PROBLEMS

of the following classic type: