recently discovered in the Shrine Library in Mashhad,
Iran, and an English edition of the text was first published in 1976.
The piece is organized as a collection of 16 discussions on original results in geometry, chiefly concerned
with the topic of CONIC SECTIONS. One sees that Diocles was the first to prove the reflection property of a
PARABOLA, thereby solving an old problem presented
by ARCHIMEDES OF SYRACUSE (ca. 287–212 B.C.E.) of
finding a mirror surface that produces heat when
placed facing the sun. (It is said that Archimedes proposed using curved mirrors to reflect the Sun’s rays and
burn the sails of enemy ships.) Diocles also describes
his “cissoid” curve in this text and a method of constructing, geometrically, the cube root of any given
length with its aid. As the construction of the cube root
of 2 is the chief stumbling block in the solution of the
duplication of the cube problem, the cissoid provides a
solution to this classic challenge. Today we describe the
cissoid as the plane curve with equation y
2
(2a – x) = x
3
,
where a is a constant. The appearance of the cube
power makes the construction of cube roots possible.
Some historians suggest Diocles may have used the
terms parabola, hyperbola, and ellipse for the conic
sections before Apollonius, the scholar usually credited
with the invention of these names. Diocles’ work on
conics greatly influenced the development of the subject. The exact date of Diocles’ death is not known.
Diophantine equation Any equation, usually in several unknowns, that is studied and required to have
only integer-valued solutions is called a Diophantine
equation. For example, the JUG-FILLING PROBLEM
requires us to find integer solutions to 3x + 5y = 1, and
the classification of PYTHAGOREAN TRIPLES seeks integer solutions to x
2 + y
2 = z
2
. These are Diophantine
problems. FERMAT’S LAST THEOREM addresses the
nonexistence of integer solutions to the generalized
equation x
n + y
n = z
n for higher-valued exponents.
Problems of this type are named after DIOPHANTUS OF
ALEXANDRIA, author of the first known book devoted
exclusively to NUMBER THEORY.
In 1900 DAVID HILBERT challenged the mathematical community to devise an ALGORITHM that would
determine whether or not any given Diophantine equation has solutions. Seventy years later Yuri Matyasevic
proved that no such algorithm can exist.
Diophantus of Alexandria (ca. 200–284 C.E.) Greek
Number theory Diophantus is remembered as the
author Arithmetica, the first known text devoted exclusively to the study of NUMBER THEORY. Ten of the original 13 volumes survive today. In considering some 130
problems, Diophantus developed general methods for
finding solutions to some surprisingly difficult integer
problems, inspiring a field of study that has since
become known as DIOPHANTINE EQUATIONs.
Essentially nothing is known about Diophantus’s
life, not even his place of birth nor the date at which he
lived. Author Metrodorus (ca. 500 C.E.), in the Greek
Anthology, briefly described the life of Diophantus
through a puzzle:
His boyhood lasted one-sixth of his life; his
beard grew after one- twelfth more; he married
after one-seventh more; and his son was born
five years later. The son lived to half his
father’s age, and the father died four years
after the son.
Setting L to be the length of Diophantus’s life, we
deduce then that the quantity:
+
+
+ 5 +
+ 4
equals the total span of his life. Setting this equal to L
and solving then yields L = 84. Of course the information provided here (that Diophantus married at age 26,
lived to age 84, and had a son who survived to age 42)
is likely fictitious. The puzzle, however, is fitting for the
type of problem Diophantus liked to solve.
In his famous text Arithmetica (Arithmetic) Diophantus presents a series of specific numerical problems, with solutions provided, that cleverly lead the
reader to an understanding of general methods and
general solutions. Diophantus ignored any solution to a
problem that was negative or involved an irrational
square root. He generally permitted only positive rational solutions. Today, going further, mathematicians call
any problem requiring only integer solutions a Diophantine equation.
Some of the problems Diophantus considered are
surprisingly difficult. For instance, in Book IV of Arithmetica Diophantus asks readers to write the number 10
as a sum of three squares each greater than three. He
provides the answer:
L
–
2
L
–
7
L
–
12
L
–
6
Diophantus of Alexandria 137
Iran, and an English edition of the text was first published in 1976.
The piece is organized as a collection of 16 discussions on original results in geometry, chiefly concerned
with the topic of CONIC SECTIONS. One sees that Diocles was the first to prove the reflection property of a
PARABOLA, thereby solving an old problem presented
by ARCHIMEDES OF SYRACUSE (ca. 287–212 B.C.E.) of
finding a mirror surface that produces heat when
placed facing the sun. (It is said that Archimedes proposed using curved mirrors to reflect the Sun’s rays and
burn the sails of enemy ships.) Diocles also describes
his “cissoid” curve in this text and a method of constructing, geometrically, the cube root of any given
length with its aid. As the construction of the cube root
of 2 is the chief stumbling block in the solution of the
duplication of the cube problem, the cissoid provides a
solution to this classic challenge. Today we describe the
cissoid as the plane curve with equation y
2
(2a – x) = x
3
,
where a is a constant. The appearance of the cube
power makes the construction of cube roots possible.
Some historians suggest Diocles may have used the
terms parabola, hyperbola, and ellipse for the conic
sections before Apollonius, the scholar usually credited
with the invention of these names. Diocles’ work on
conics greatly influenced the development of the subject. The exact date of Diocles’ death is not known.
Diophantine equation Any equation, usually in several unknowns, that is studied and required to have
only integer-valued solutions is called a Diophantine
equation. For example, the JUG-FILLING PROBLEM
requires us to find integer solutions to 3x + 5y = 1, and
the classification of PYTHAGOREAN TRIPLES seeks integer solutions to x
2 + y
2 = z
2
. These are Diophantine
problems. FERMAT’S LAST THEOREM addresses the
nonexistence of integer solutions to the generalized
equation x
n + y
n = z
n for higher-valued exponents.
Problems of this type are named after DIOPHANTUS OF
ALEXANDRIA, author of the first known book devoted
exclusively to NUMBER THEORY.
In 1900 DAVID HILBERT challenged the mathematical community to devise an ALGORITHM that would
determine whether or not any given Diophantine equation has solutions. Seventy years later Yuri Matyasevic
proved that no such algorithm can exist.
Diophantus of Alexandria (ca. 200–284 C.E.) Greek
Number theory Diophantus is remembered as the
author Arithmetica, the first known text devoted exclusively to the study of NUMBER THEORY. Ten of the original 13 volumes survive today. In considering some 130
problems, Diophantus developed general methods for
finding solutions to some surprisingly difficult integer
problems, inspiring a field of study that has since
become known as DIOPHANTINE EQUATIONs.
Essentially nothing is known about Diophantus’s
life, not even his place of birth nor the date at which he
lived. Author Metrodorus (ca. 500 C.E.), in the Greek
Anthology, briefly described the life of Diophantus
through a puzzle:
His boyhood lasted one-sixth of his life; his
beard grew after one- twelfth more; he married
after one-seventh more; and his son was born
five years later. The son lived to half his
father’s age, and the father died four years
after the son.
Setting L to be the length of Diophantus’s life, we
deduce then that the quantity:
+
+
+ 5 +
+ 4
equals the total span of his life. Setting this equal to L
and solving then yields L = 84. Of course the information provided here (that Diophantus married at age 26,
lived to age 84, and had a son who survived to age 42)
is likely fictitious. The puzzle, however, is fitting for the
type of problem Diophantus liked to solve.
In his famous text Arithmetica (Arithmetic) Diophantus presents a series of specific numerical problems, with solutions provided, that cleverly lead the
reader to an understanding of general methods and
general solutions. Diophantus ignored any solution to a
problem that was negative or involved an irrational
square root. He generally permitted only positive rational solutions. Today, going further, mathematicians call
any problem requiring only integer solutions a Diophantine equation.
Some of the problems Diophantus considered are
surprisingly difficult. For instance, in Book IV of Arithmetica Diophantus asks readers to write the number 10
as a sum of three squares each greater than three. He
provides the answer:
L
–
2
L
–
7
L
–
12
L
–
6
Diophantus of Alexandria 137