308 Legendre, Adrien-Marie

Gottfried Wilhelm Leibniz, an eminent mathematician of the 18th

century, was the first to found and publish an account of differen-

tial calculus. He later developed the theory of integral calculus

and the fundamental theorem of calculus that connects the two

subjects. (Photo courtesy of Feltz/Topham/The Image Works)

function f(x) = x+ |x|, for example, at the position

The left derivative (with hnegative)

is where

as the right derivative (with hpositive) is

. The tan-

gent line to the curve has slope zero just to the left of

x= 0 and slope 2 just to its right. This inconsistency

shows that there is no well-defined tangent line to the

graph f(x) = x+ |x| at x= 0.

Legendre, Adrien-Marie (1752–1833) French Geom-

etry, Number theory Born on September 18, 1752, in

Paris, France, Adrien-Marie Legendre is remembered as

a capable French mathematician who devoted his life to

the subject but had the unfortunate luck to see the bulk

of his work rendered obsolete by the discoveries and

abilities of younger, brighter mathematicians.

Legendre published in the fields of

NUMBER THEORY

,

elliptic functions, E

UCLIDEAN GEOMETRY

, and celestial

mechanics. His name is associated with a number of

mathematical concepts, including Legendre polynomials,

an important class of functions useful for solving certain

types of

DIFFERENTIAL EQUATION

s; the

LEAST SQUARES

METHOD

; and the Legendre symbol in number theory.

He published an elementary geometry text, Eléments de

géométrie (Elements of geometry), which dominated the

teaching of geometry in America and Europe throughout

the 19th century. In 1791 the Académie des Sciences

undertook the task to standardize weights and measures

and directed Legendre to make the necessary astronomi-

cal observations to compute the length of a meter.

Little is known of Legendre’s early life. Born into a

family of wealth, Legendre had access to an excellent

formal education and the means to pursue scholarly

interests. In 1770 Legendre defended a thesis in mathe-

matics and physics at the Collège Mazarin. His 1782

mathematical research on the path of projectiles earned

him an award from the Berlin Academy and garnered

him some attention as a scholar of mathematics. A year

later Legendre was appointed an adjoint position at the

Académie des Sciences in Paris, later renamed the Insti-

tut National des Sciences et des Arts. He remained

there until 1824, rising appropriately in rank and posi-

tion throughout the decades. However, due to a politi-

cal disagreement with the running of the Institut,

Legendre was denied a pension in his retirement and

died in poverty 9 years later on January 9, 1833.

Despite his work being seen as obsolete, Legendre

did raise a number of fundamental questions in the

fields of number theory and elliptical function theory

that spurred a great deal of mathematical investigation

during the century that followed him.

Leibniz, Gottfried Wilhelm (1646–1716) German

Calculus, Logic Born on July 1, 1646, in Leipzig, Sax-

ony (now Germany), scholar Gottfried Wilhelm Leibniz

is remembered for discovering, and being the first to pub-

lish in 1684, the theory of

DIFFERENTIAL CALCULUS

. In

subsequent years he also developed the theory of

INTE

-

GRAL CALCULUS

and formulated the

FUNDAMENTAL THE

-

OREM OF CALCULUS

that unites the two fields. This work

was accomplished independently of the progress made by