recently discovered in the Shrine Library in Mashhad,
Iran, and an English edition of the text was first pub-
lished in 1976.
The piece is organized as a collection of 16 discus-
sions on original results in geometry, chiefly concerned
with the topic of
CONIC SECTIONS
. One sees that Dio-
cles was the first to prove the reflection property of a
PARABOLA
, thereby solving an old problem presented
by A
RCHIMEDES OF
S
YRACUSE
(ca. 287–212
B
.
C
.
E
.) of
finding a mirror surface that produces heat when
placed facing the sun. (It is said that Archimedes pro-
posed 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 con-
structing, 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 y2(2a– x) = x3,
where ais 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 sub-
ject. The exact date of Diocles’ death is not known.
Diophantine equation Any equation, usually in sev-
eral 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 inte-
ger solutions to x2+ y2= z2. These are Diophantine
problems. F
ERMAT
’
S LAST THEOREM
addresses the
nonexistence of integer solutions to the generalized
equation xn+ yn= znfor higher-valued exponents.
Problems of this type are named after D
IOPHANTUS OF
A
LEXANDRIA
, author of the first known book devoted
exclusively to
NUMBER THEORY
.
In 1900 D
AVID
H
ILBERT
challenged the mathemati-
cal community to devise an
ALGORITHM
that would
determine whether or not any given Diophantine equa-
tion 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 exclu-
sively to the study of
NUMBER THEORY
. Ten of the orig-
inal 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 D
IOPHANTINE EQUATION
s.
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 Lto 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 informa-
tion 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) Dio-
phantus presents a series of specific numerical prob-
lems, 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 ratio-
nal solutions. Today, going further, mathematicians call
any problem requiring only integer solutions a Dio-
phantine equation.
Some of the problems Diophantus considered are
surprisingly difficult. For instance, in Book IV of Arith-
metica 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