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.
ARCHIMEDES OF SYRACUSE (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 ARCHIMEDEAN 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.)
APOLLONIUS OF PERGA (ca. 262–190 B.C.E.) continued work on conic sections and is credited for properly 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 astronomy, trigonometry, and algebra, with just a few
notable exceptions.
In the second century C.E., Greek astronomer
CLAUDIUS PTOLEMY corrected and extended Hipparchus’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 proving Euclid’s PARALLEL POSTULATE. In the third century,
DIOPHANTUS OF ALEXANDRIA produced his famous
text Arithmetica (Arithmetic), from which the study
of DIOPHANTINE EQUATIONs was born. In the midfourth century, the enthusiastic PAPPUS OF ALEXANDRIA attempted to revive interest in ardent
mathematical research of the Greek style. He produced 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 succeed in his general goal. After Pappus, of note is
HYPATIA OF ALEXANDRIA (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 PROCLUS (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 ARCHYTAS OF TARENTUM; DEDUCTIVE/
INDUCTIVE REASONING; ERATOSTHENES OF CYRENE;
EUCLID’S POSTULATES; EUCLIDEAN ALGORITHM EUCLIDEAN GEOMETRY; EUDOXUS OF CNIDUS; HERON OF
ALEXANDRIA; HIPPASUS OF METAPONTUM; HIPPOCRATES OF CHIOS; HISTORY OF EQUATONS AND ALGEBRA; MENELAUS OF ALEXANDRIA; PAPPUS’S THEOREMS;
PTOLEMY’S THEOREM; PYTHAGOREAN TRIPLES; THEODORUS OF CYRENE; ZENO’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 mathematical 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 electromagnetism, important for physicists, Green also established a fundamental mathematical technique for
computing CONTOUR INTEGRALs and DOUBLE INTEGRALs. The famous theorem that bears his name states
that if a region R in the xy-plane is bounded by a curve
C, and if functions P(x,y) and Q(x,y) have continuous
PARTIAL DERIVATIVEs, then:
This result appears in every multivariable calculus textbook of today.
After reading the famous 1828 piece, mathematician Sir Edward Bromhead invited Green to continue
P x y dx Q x y dy
Q
x
P
y
dA
R
C
( , )
( , )
+
=
∂
∂
−
∂
∂
∫∫
∫
Green, George 239
ANGLE and DUPLICATING THE CUBE, spurred a great
deal of significant further research in mathematics for
centuries to come.
ARCHIMEDES OF SYRACUSE (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 ARCHIMEDEAN 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.)
APOLLONIUS OF PERGA (ca. 262–190 B.C.E.) continued work on conic sections and is credited for properly 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 astronomy, trigonometry, and algebra, with just a few
notable exceptions.
In the second century C.E., Greek astronomer
CLAUDIUS PTOLEMY corrected and extended Hipparchus’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 proving Euclid’s PARALLEL POSTULATE. In the third century,
DIOPHANTUS OF ALEXANDRIA produced his famous
text Arithmetica (Arithmetic), from which the study
of DIOPHANTINE EQUATIONs was born. In the midfourth century, the enthusiastic PAPPUS OF ALEXANDRIA attempted to revive interest in ardent
mathematical research of the Greek style. He produced 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 succeed in his general goal. After Pappus, of note is
HYPATIA OF ALEXANDRIA (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 PROCLUS (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 ARCHYTAS OF TARENTUM; DEDUCTIVE/
INDUCTIVE REASONING; ERATOSTHENES OF CYRENE;
EUCLID’S POSTULATES; EUCLIDEAN ALGORITHM EUCLIDEAN GEOMETRY; EUDOXUS OF CNIDUS; HERON OF
ALEXANDRIA; HIPPASUS OF METAPONTUM; HIPPOCRATES OF CHIOS; HISTORY OF EQUATONS AND ALGEBRA; MENELAUS OF ALEXANDRIA; PAPPUS’S THEOREMS;
PTOLEMY’S THEOREM; PYTHAGOREAN TRIPLES; THEODORUS OF CYRENE; ZENO’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 mathematical 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 electromagnetism, important for physicists, Green also established a fundamental mathematical technique for
computing CONTOUR INTEGRALs and DOUBLE INTEGRALs. The famous theorem that bears his name states
that if a region R in the xy-plane is bounded by a curve
C, and if functions P(x,y) and Q(x,y) have continuous
PARTIAL DERIVATIVEs, then:
This result appears in every multivariable calculus textbook of today.
After reading the famous 1828 piece, mathematician Sir Edward Bromhead invited Green to continue
P x y dx Q x y dy
Q
x
P
y
dA
R
C
( , )
( , )
+
=
∂
∂
−
∂
∂
∫∫
∫
Green, George 239