318 Liu Hui

is transcendental.

Liouville died in Paris, France, on September 8,

1882. He established his place in the history of mathe-

matics for pioneering work on the study of

REAL

NUMBERS

.

Liu Hui See C

HINESE MATHEMATICS

.

Li Ye (Li Chi, Li Zhi) (1192–1279) Chinese Algebra

Li Ye is remembered for his 1248 text Ceyuan Haijing

(Sea mirror of circle measurements), in which he intro-

duced the “method of the celestial element”—a system

of notation for polynomials in one variable (the celes-

tial element)—and a set of techniques for solving such

polynomial equations. Using the single diagram of a

circular city wall inscribed inside a large right-angled

triangle, Ceyuan Haijing leads the reader through 170

geometric problems, cleverly designed to illustrate the

techniques of translating geometry into algebra, and

then solving the consequent algebraic equations. Over

650 different formulae for triangular areas and seg-

ment lengths are presented in this text.

Extremely little is known of Li Ye’s life, except that

his work apparently earned him some regard. In 1266 Li

Ye was appointed a position at the elite Hanlin Academy

by the emperor Kublai Khan, grandson of the great

Genghis Khan. He remained there only a few years to

then retire and live the rest of his life as a hermit.

Lobachevsky, Nikolai Ivanovich (1792–1856) Rus-

sian Geometry Born on December 1, 1792, in Nov-

gorod, Russia, Nikolai Lobachevsky is remembered for

his 1826 discovery of

HYPERBOLIC GEOMETRY

and for

detailing many of its properties. (This work was con-

ducted independently of the discoveries made by J

ÁNOS

B

OLYAI

(1802–60) 3 years earlier.) He was the first to

publish a description of a

NON

-E

UCLIDEAN GEOMETRY

in his 1826 article “A Concise Outline of the Founda-

tions of Geometry.”

Lobachevsky entered Kazan State University in 1807

with the intent to study medicine, but soon changed

interest to pursue courses in mathematics and physics.

He graduated with a master’s degree in 1811 and, three

years later, was appointed a lectureship position at the

university. In 1822 he was appointed full professor, and

in 1827 was named rector of the university. He remained

at Kazan State University until his retirement in 1846.

Lobachevsky was introduced to the topic of geome-

try as a student. Ever since the time of E

UCLID

(ca.

300–260

B

.

C

.

E

.), scholars questioned Euclid’s choice of

axioms as the basis for all of geometry. His fifth postu-

late, the famous

PARALLEL POSTULATE

, was deemed of a

different nature than the remaining four, and scholars

suspected that it could be deduced from them as a logi-

cal consequence. This became an outstanding challenge

in mathematics. For two millennia scholars attempted

to establish the fifth postulate as a

THEOREM

, but failed.

Upon learning of this problem, Lobachevsky began

analyzing the situation for himself. Rather than attempt

to prove the fifth postulate, he considered the possibil-

ity that it need not follow from the remaining four

axioms, and, moreover, allowed for the possibility of a

geometry in which the first four axioms do hold but

one in which the fifth postulate is blatantly false. With

this expanded thinking, Lobachevsky discovered a con-

sistent theory of geometry—hyperbolic geometry—dif-

ferent from Euclidean geometry, but nonetheless valid

in its own right. That such a geometry exists showed,

once and for all, that the fifth postulate is in fact inde-

pendent of the remaining four axioms. This was a

remarkable achievement.

Lobachevsky presented the results of his discovery

to his colleagues at Kazan State University in 1826

and published his article “A Concise Outline of the

Foundations of Geometry” in the Kazan Messenger.

Unfortunately, the St. Petersburg Academy of Sciences

decided not to publish his piece as a peer-reviewed

article, and Lobachevsky’s work did not receive

widespread recognition.

Although Lobachevsky managed to publish

papers on the topic at later dates, including his 1840

German paper “Geometrische Untersuchnungen zur

Theorie der Parallellinien” (Geometric investigation

on the theory of parallel lines), it was not until after

his death on February 24, 1856, that the importance

of his work was understood and published in a main-

stream forum. Today, Lobachevskian geometry plays a

central role in the modern description of space and

motion in relativistic quantum mechanics and the gen-

eral theory of relativity.