his methods of algebra paved the way for significant
developments in mathematics of much broader scope
than ever previously envisioned.
The mathematician al-Mahani (ca. 820) developed
refined approaches for reducing geometric problems to
algebraic ones. He showed, in particular, that the famous
problem of
DUPLICATING THE CUBE
is essentially an alge-
braic issue. Other scholars brought rigor to the subject
by proving that certain popular, but complicated, alge-
braic methods were valid. These scholars were comfort-
able manipulating
POLYNOMIAL
s and developed rules for
working with
EXPONENT
s, They solved linear and
QUADRATIC
equations, as well as various
SYSTEM
s
OF
EQUATIONS
. Surprisingly, no one of the time thought to
ease matters by using symbols to represent quantities: all
equations and all manipulations were described fully in
words each and every time they were employed.
With quadratic equations well understood, the
scholar O
MAR
K
HAYYAM
(ca. 1048–1131) attempted to
develop methods of solving degree-three equations.
Although he was unable to develop general algebraic
methods for this task, he did find ingenious geometric
techniques for solving certain types of cubics with the
aid of
CONIC SECTIONS
. He was aware that such equa-
tions could have more than one solution.
In number theory, Thabit ibn Qurra (ca. 836–901)
found a beautiful method for generating
AMICABLE NUM
-
BERS
. This technique was later utilized by al-Farisi
(ca. 1260–1320) to yield the pair 17,296 and 18,416,
which today is usually attributed to L
EONHARD
E
ULER
(1707–83). In his writing, Omar Khayyam referred to
earlier Arab texts, now lost, that discuss the equivalent
of P
ASCAL
’s
TRIANGLE
and its connections to the
BINOMIAL THEOREM
. The mathematician al-Haytham
(ca. 965–1040) attempted to classify all even
PERFECT
NUMBER
s.
Taking advantage of the ease of the Indian system
of decimal place-value representation, Arabic scholars
also made great advances in numeric computations.
The great 14th-century scholar J
AMSHID AL
-K
ASHI
developed effective methods for extracting the nth root
of a number, and evaluated πto a significant number of
decimal places. Scholars at the time also developed
effective methods for computing trigonometric tables
and techniques for making highly accurate computa-
tions for the purposes of astronomy.
On a theoretical note, scholars also advanced the
general understanding of trigonometry and explored
problems in spherical geometry. They also investigated
the philosophical underpinnings of geometry, focusing,
in particular, on the role the famous
PARALLEL POSTU
-
LATE
plays in the theory. Omar Khayyam, for instance,
attempted to prove the parallel postulate—failing, of
course—but did accidentally prove results about figures
in non-Euclidean geometries along the way. The mathe-
matician Ibrahim ibn Sinan (908–946) also introduced
a method of “integration” for calculating volumes and
areas following an approach more general than that
developed by Archimedes of Syracuse (ca. 287–212
B
.
C
.
E
.). He also applied his approach to the study of
CONIC SECTIONS
and to optics.
See also
BASE OF A NUMBER SYSTEM
;
HISTORY OF
EQUATIONS AND ALGEBRA
(essay);
HISTORY OF GEOME
-
TRY
(essay);
HISTORY OF TRIGONOMETRY
(essay).
arc Part of a continuous curve between two given
points on the curve is called an arc of the curve. In par-
ticular, two points on a
CIRCLE
determine two arcs. If the
circumference of the circle is divided by them into two
unequal parts, then the smaller portion is usually called
the minor arc of the circle and the larger the major arc.
Archimedean spiral See
SPIRAL OF
A
RCHIMEDES
.
Archimedes of Syracuse (287–212
B
.
C
.
E
.) Greek
Geometry, Mechanics Born in the Greek colony of
Syracuse in Sicily, Archimedes is considered one of the
greatest mathematicians of all time. He made consider-
able contributions to the fields of planar and solid
GEOMETRY
, hydrostatics, and mechanics. In his works
Measurement of a Circle and Quadrature of the Para-
bola, Archimedes solved difficult problems of mensu-
ration in planar geometry by inventing an early
technique of
INTEGRAL CALCULUS
, which he called the
“method of exhaustion.” This allowed him to compute
areas and lengths of certain curved figures. Later, in his
works On the Sphere and Cylinder and On Conoids
and Spheroids, he applied this technique to also com-
pute the volume and surface area of the sphere and
other solid objects. In his highly original work On
Floating Bodies, Archimedes developed the mathemat-
ics of hydrostatics and equilibrium, along with the
LAW
OF THE LEVER
, the notion of specific gravity, and tech-
niques for computing the
CENTER OF GRAVITY
of a
18 arc