√a12+ a22

√(b1– a1)2+ (b2– a2)2

310 L’Hôpital, Guillaume François Antoine, marquis de

to our concept of one “mile.” One-thousandth of a foot

is called a “gry.” Lengths defined by body measure-

ments are subject to great variation, and it was found

that different communities throughout the world were

using different standards of length even though the

units used were given the same name. During the 19th

and 20th centuries an international standard of units

was developed, and unambiguous units of length were

defined.

In mathematics, the notion of length appears in sev-

eral settings and can be given precise definitions. For

example, on the number line, the length of a line seg-

ment connecting a number ato number bis given by

the

ABSOLUTE VALUE

|b– a|; the length of a line segment

connecting points A= (a1,a2) and B= (b1,b2) is given

by the

DISTANCE FORMULA

; and

the length of a

VECTOR

a= <a1,a2> is given by



a



=

. (These latter formulae generalize to three

and higher dimensions.)

CALCULUS

can be used to find the

ARC LENGTH

of

curved lines in two- and three-dimensional space.

See also

SI UNITS

.

L’Hôpital, Guillaume François Antoine, marquis de

(1661–1704) French Calculus Born in Paris, France,

in 1661 (his exact birth date is not known), Guillaume

l’Hôpital is remembered as the famed author of the

first textbook on the topic of

DIFFERENTIAL CALCULUS

,

Analyse des infiniment petits (Analysis with infinitely

small quantities), written in 1696. Apart from explain-

ing the methods and details of the newly discovered

theory, this work also contains the first formulation of

the rule that now bears his name.

L’Hôpital’s talent for mathematics was recognized

as a boy. At the age of 15 he solved a problem on the

CYCLOID

put forward by B

LAISE

P

ASCAL

(1623–62) and

later contributed to the solution of the famous

BRACHISTOCHRONE

problem. Before pursuing mathe-

matics in earnest, l’Hôpital served as a cavalry officer

but soon had to resign due to nearsightedness.

In 1691 l’Hôpital hired Swiss mathematician

Johann Bernoulli of the B

ERNOULLI FAMILY

to teach

him the newly discovered theory of calculus. This was

conducted chiefly by correspondence, and the agree-

ment was made that all content of the letters sent

between them would belong to the marquis. This mate-

rial formed the basis of his 1696 text.

Chapter one of Analyse des infiniment petits

defines the notion of a

DIFFERENTIAL

(or difference, as

l’Hôpital called it) and provides rules as to how they

are to be manipulated. It also outlines the basic princi-

ples of differential calculus. The second chapter gives

the method for determining the tangent line to a curve,

and chapter three deals with

MAXIMUM

/

MINIMUM

problems using problems from mechanics and geogra-

phy as examples. Later chapters deal with cusps, points

of inflection, higher-order derivatives, evolutes, and

caustics. L’H

ÔPITAL

’

S RULE

appears in chapter nine.

After l’Hôpital’s death Johann Bernoulli com-

plained publicly that not enough credit was given to

him for the work contained in the text. (L’Hôpital did

write a note of gratitude in the book to Bernoulli, and

to G

OTTFRIED

W

ILHELM

L

EIBNIZ

, for contributing their

ideas.) It is known today, for example, that Bernoulli,

not l’Hôpital, discovered l’Hôpital’s rule.

L’Hôpital wrote a complete manuscript for a sec-

ond book Traité analytique des sections coniques

(Analytical treatise on conic sections), which was pub-

lished posthumously in 1720. He had also planned to

write a third text, one on the topic of integral calcu-

lus, but discontinued work on the project when he

heard that Leibniz was working on his own book on

the topic.

L’Hôpital died in Paris, France, on February 2,

1704.

L’Hôpital’s rule (L’Hospital’s rule) Named after G

UIL

-

LAUME

F

RANÇOIS

A

NTOINE L

’H

ÔPITAL

(1661–1704), a

student of the mathematician Johann Bernoulli of the

B

ERNOULLI FAMILY

, l’Hôpital’s rule is a method for

finding the

LIMIT

of a ratio of two functions, each of

which separately tends to zero. Precisely:

Suppose f(x) and g(x) are two differentiable

functions with f(a) = 0 and g(a) = 0 at some

point a. Then the limit of the ratio f(x)/g(x) as

x→ais equal to the limit of the ratio of the

derivatives f′(x)/g′(x) as x→a(provided the

derivative of g(x) is never zero, except possibly

at x= a).

As an example, to compute the limit

(which looks to be of the form 0/0), one simply takes the

derivative of numerator and denominator separately: