Egyptians of the time held mathematics in high regard.

The copyist, whose name is rendered as Ahmes or

Ahmose, copied also the grandiose title of the papyrus:

Accurate Rendering: The Entrance into the Knowledge

of All Existing Things and All Obscure Secrets. The

purpose of the text seems to be that of a textbook—a

series of examples and exercises organized to illustrate

certain mathematical skills and techniques. Some of the

latter problems discussed, however, have no practical

application and clearly indicate a delight in studying

mathematics for its own sake.

See also E

GYPTIAN FRACTIONS

; E

GYPTIAN MATHE

-

MATICS

; E

GYPTIAN MULTIPLICATION

.

Richter scale See

LOGARITHMIC SCALE

.

Riemann, Georg Friedrich Bernhard (1826–1866)

German Geometry, Analysis, Calculus, Number Theory

Born on September 17, 1826, in Breselenz, Hanover

(now Germany), Bernhard Riemann is remembered as

one of the greatest mathematicians of the 19th century.

His work in

NON

-E

UCLIDEAN GEOMETRY

revolutionized

modern thinking on the study of shape and space and

provided the tools A

LBERT

E

INSTEIN

needed for his

description of the universe in the general theory of rela-

tivity. Among his many significant contributions to

ANALYSIS

, Riemann is remembered for his construction

of the Riemann integral, which, for the first time,

extended the theory of

CALCULUS

beyond the realm of

continuous functions only. Riemann also developed a

theory of complex functions (that is, functions whose

inputs and outputs are

COMPLEX NUMBERS

) and studied

applications of calculus in this setting. In 1859, while

searching for a better approximation for the number of

primes as given by the famous

PRIME

-

NUMBER THEOREM

,

Riemann discovered a generalized version of L

EON

-

HARD

E

ULER

’s important

ZETA FUNCTION

. He made a

deep and insightful conjecture about the properties of

this function, the Riemann hypothesis, which, if true,

would provide invaluable information about the distri-

bution of prime numbers. Proving (or disproving) the

Riemann hypothesis is considered the most important

unsolved mathematical problem of today.

Riemann exhibited a keen interest in mathematics

at an early age and was encouraged by the director of

his high school to pursue this line of study. In 1846

Riemann enrolled at the University of Göttingen to

study under the mathematician C

ARL

F

RIEDRICH

G

AUSS

(1777–1855). Surprisingly, even with Gauss as a faculty

member, Göttingen did not offer a strong program in

mathematics, and so, a year later, Riemann transferred

to Berlin University, an institution recognized for its

strength in mathematics at the time. While there, Rie-

mann developed his general theory of complex vari-

ables and functions that later formed the basis of much

of his important work.

After completing his undergraduate degree at

Berlin, Riemann returned to Göttingen in 1849 to com-

plete a doctoral thesis under the supervision of Gauss.

He was awarded a Ph.D. 2 years later in mathematics.

Gauss described Riemann’s thesis work as having

“a gloriously fertile originality.” By viewing the range

of outputs of a complex function as a surface in three-

or even four-dimensional space, Riemann managed to

tie the theory of calculus and analysis to the study of

shape, space, and geometry. This innovative approach

was immediately recognized as a significant turning

point in the advancement of mathematics. On Gauss’s

recommendation, Riemann was appointed a position at

Göttingen, and he commenced work on his habilitation

degree. During this time Riemann turned his attention

to the study of F

OURIER SERIES

and questions of inte-

grability. It was during this period that Riemann for-

mulated his famous definition of an integral.

As the final step to completing his degree Riemann

was required to give a public lecture. Gauss asked him

to speak on the theory of geometry, and on June 10,

1854, Riemann presented to the outside world his

groundbreaking work on shape and space.

Riemann devised a general notion of distance in n-

dimensional space and developed a theory of

TENSOR

s

that allowed him to properly define geometric shape. In

particular, Riemann developed the notion of curvature

tensor to measure, or define, the warping of space. The

mathematics Riemann presented on that day proved to

be the precise framework Einstein needed for structur-

ing his theory of general relativity. Riemann continued

to present and publish groundbreaking work in the

years that followed.

On July 30, 1859, Riemann was appointed chair

of mathematics at Göttingen, and a few days later he

was honored with election to the Berlin Academy of

Sciences. By this time Riemann had developed an inter-

est in

NUMBER THEORY

and the distribution of prime

Riemann, Georg Friedrich Bernhard 447