Green, George 239

challenge, as well as the problems of

TRISECTING AN

ANGLE

and

DUPLICATING THE CUBE

, spurred a great

deal of significant further research in mathematics for

centuries to come.

A

RCHIMEDES OF

S

YRACUSE

(ca. 287–212

B

.

C

.

E

.)

solved the problem of squaring the parabola, as well as

made significant advances in computing the areas and

volumes of curved figures and solids. (He also

“solved” the problem of squaring the circle by making

use of his A

RCHIMEDEAN SPIRAL

. Unfortunately, his

method went beyond the use of a straightedge and

compass alone, and so is not a permissible solution to

the original problem.)

A

POLLONIUS OF

P

ERGA

(ca. 262–190

B

.

C

.

E

.) con-

tinued work on conic sections and is credited for prop-

erly defining an

ELLIPSE

,a

HYPERBOLA

, and a

PARABOLA

. Around the same time, Greek astronomer

Hipparchus wrote a table of “chord values” (the

equivalent to a modern table of sine values), which he

used to solve astronomical problems. This represented

the beginning development of

TRIGONOMETRY

in

Greek mathematics, but also marked an end of fervent

mathematical development in the Greek tradition. For

the five centuries that followed, new developments

were limited to straightforward advances in astron-

omy, trigonometry, and algebra, with just a few

notable exceptions.

In the second century

C

.

E

., Greek astronomer

C

LAUDIUS

P

TOLEMY

corrected and extended Hip-

parchus’s table and clarified the mathematics that is

used to produce such a table. He is also known as one

of the first scholars to make a serious attempt at prov-

ing Euclid’s

PARALLEL POSTULATE

. In the third century,

D

IOPHANTUS OF

A

LEXANDRIA

produced his famous

text Arithmetica (Arithmetic), from which the study

of D

IOPHANTINE EQUATION

s was born. In the mid-

fourth century, the enthusiastic P

APPUS OF

A

LEX

-

ANDRIA

attempted to revive interest in ardent

mathematical research of the Greek style. He pro-

duced his treatise Synagoge (Collections) to act as a

commentary and guide to all the geometric works of

his time and included in it a significant number of

original results, extensions of ideas, and innovative

shifts of perspective. Unfortunately, he did not suc-

ceed in his general goal. After Pappus, of note is

H

YPATIA OF

A

LEXANDRIA

(370–415), the first woman

to be named in the history of mathematics, credited

for writing insightful commentaries on the works of

Apollonius and Diophantus, and P

ROCLUS

(ca.

410–485), who is noted for his detailed commentary

on the work of Euclid and his own attempt to prove

the parallel postulate.

The beginning of the fifth century marks a clear

end to the tradition of Greek mathematics.

See also A

RCHYTAS OF

T

ARENTUM

;

DEDUCTIVE

/

INDUCTIVE REASONING

; E

RATOSTHENES OF

C

YRENE

;

E

UCLID

’

S POSTULATES

; E

UCLIDEAN ALGORITHM

E

UCLI

-

DEAN GEOMETRY

; E

UDOXUS OF

C

NIDUS

; H

ERON OF

A

LEXANDRIA

; H

IPPASUS OF

M

ETAPONTUM

; H

IPPO

-

CRATES OF

C

HIOS

;

HISTORY OF EQUATONS AND ALGE

-

BRA

; M

ENELAUS OF

A

LEXANDRIA

; P

APPUS

’

S THEOREMS

;

P

TOLEMY

’

S THEOREM

; P

YTHAGOREAN TRIPLES

; T

HEO

-

DORUS OF

C

YRENE

; Z

ENO

’

S PARADOXES

.

Green, George (1793–1841) British Calculus Born

in July 1793 (his exact birth date is not known) in

Nottingham, England, George Green is remembered

today for his influential 1828 paper “Essay on the

Application of Mathematical Analysis to the Theory of

Electricity and Magnetism,” in which he developed the

notion of “potential” and proved a fundamental math-

ematical result today known as Green’s theorem.

Green left school at the age of nine and worked in

his father’s bakery for the next 30 years. Historians do

not know how Green developed an understanding of

mathematics, nor how he had access to current work in

the field, but in 1828 he published one of the most

important scientific papers of the time. Apart from

advancing the mathematical understanding of electro-

magnetism, important for physicists, Green also estab-

lished a fundamental mathematical technique for

computing

CONTOUR INTEGRAL

s and

DOUBLE INTE

-

GRAL

s. The famous theorem that bears his name states

that if a region Rin the xy-plane is bounded by a curve

C, and if functions P(x,y) and Q(x,y) have continuous

PARTIAL DERIVATIVE

s, then:

This result appears in every multivariable calculus text-

book of today.

After reading the famous 1828 piece, mathemati-

cian Sir Edward Bromhead invited Green to continue