CALCULUS

forced scholars to seek greater levels of rigor

and abstraction. Even the nature of logical reasoning

itself was examined as an attempt to understand and

resolve fundamental paradoxes. The need for abstract

ANALYSIS

and synthesis was recognized as important,

and dichotomy between applied and pure mathematics

became more apparent.

Today many research universities possess two

departments of mathematics, one considered pure and

the other applied. Students can obtain advanced degrees

in either field.

See also B

ABYLONIAN MATHEMATICS

; G

REEK

MATHEMATICS

.

pyramid A

CONE

with a polygon for the base is

called a pyramid. Specifically, a pyramid is the solid fig-

ure formed by a polygon and a number of triangles,

one attached to each side of the polygon, all meeting at

a common point called the

APEX

of the pyramid. The

pyramid is classified as a right pyramid if the line con-

necting the apex of the pyramid to the

CENTER OF

GRAVITY

of the base meets the base at a right angle.

(Otherwise it is called an oblique pyramid.) A square

pyramid is a right pyramid with a square base.

The study of volumes of cones shows that the vol-

ume of a pyramid is just one-third the area of the base

of the figure times its height. A

TETRAHEDRON

is a

right pyramid with four equilateral triangles as faces.

See also

FRUSTUM

.

Pythagoras (ca. 569–475

B

.

C

.

E

.) Greek Geometry,

Number theory Born in Samos, Ionia, Greek mathe-

matician and mystic Pythagoras is remembered as

founder of the Pythagorean School, which claimed to

have found universal truth through the study of num-

ber. As such, the Pythagoreans are credited as the first

to have taken mathematics seriously as a study in its

own right without regard to possible application and

practical need. What precisely Pythagoras con-

tributed himself is no longer clear, but the school is

acknowledged to have discovered the role of ratios of

whole numbers in the musical scale, the properties of

FIGURATE NUMBERS

,

AMICABLE NUMBERS

,

PERFECT

NUMBER

s, and the existence of

IRRATIONAL NUMBERS

.

Although P

YTHAGORAS

’

S THEOREM

was known to the

Babylonians over 1,000 years earlier, the Pythagore-

ans are the first to have provided a general proof of

the result.

Very little is known of Pythagoras’s life. The sect he

founded was half scientific and half religious and fol-

lowed a code of secrecy that certainly promoted great

mystery about the man himself. Many historical writ-

ings attribute godlike qualities to Pythagoras and are

generally not regarded as accurate portrayals. It is

understood that Pythagoras visited with T

HALES OF

M

ILETUS

(ca. 625–547

B

.

C

.

E

.) as a young man who,

most likely, contributed to Pythagoras’s interest in

mathematics. Around 535

B

.

C

.

E

., Pythagoras traveled

to Egypt and was certainly influenced by the secret

sects of the Egyptian priests. (Many of the practices the

Pythagoreans followed were the same as those prac-

ticed in Egypt—refusal to eat beans or to wear cloth

made from animal skins, for instance.)

In 525

B

.

C

.

E

., Egypt was invaded by Persian forces,

and Pythagoras was captured and taken to Babylon.

Pythagoras 423

Pythagoras, leader of a sixth-century

B

.

C

.

E

. sect, claimed to have

found universal truth through the study of whole numbers and

their ratios. (Photo courtesy of Topham/The Image Works)