to our concept of one “mile.” One-thousandth of a foot
is called a “gry.” Lengths defined by body measurements 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 several settings and can be given precise definitions. For
example, on the number line, the length of a line segment connecting a number a to number b is given by
the ABSOLUTE VALUE |b – a|; the length of a line segment
connecting points A = (a 1 ,a 2 ) and B = (b 1 ,b 2 ) is given
by the DISTANCE FORMULA
; and
the length of a VECTOR a = is given by ʈaʈ =
. (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 explaining 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 BLAISE PASCAL (1623–62) and
later contributed to the solution of the famous
BRACHISTOCHRONE problem. Before pursuing mathematics 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 BERNOULLI FAMILY to teach
him the newly discovered theory of calculus. This was
conducted chiefly by correspondence, and the agreement was made that all content of the letters sent
between them would belong to the marquis. This material 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 principles 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 geography as examples. Later chapters deal with cusps, points
of inflection, higher-order derivatives, evolutes, and
caustics. L’HÔPITAL’S RULE appears in chapter nine.
After l’Hôpital’s death Johann Bernoulli complained 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 GOTTFRIED WILHELM LEIBNIZ, 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 second book Traité analytique des sections coniques
(Analytical treatise on conic sections), which was published posthumously in 1720. He had also planned to
write a third text, one on the topic of integral calculus, 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 GUILLAUME FRANÇOIS ANTOINE L’HÔPITAL (1661–1704), a
student of the mathematician Johann Bernoulli of the
BERNOULLI 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 → a is 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:
lim
x
x
x
→
−
−
1
2 1
2
2
√a 1
2 + a 2
2
√(b 1 – a 1 )
2 + (b 2 – a 2 )
2
310 L’Hôpital, Guillaume François Antoine, marquis de
is called a “gry.” Lengths defined by body measurements 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 several settings and can be given precise definitions. For
example, on the number line, the length of a line segment connecting a number a to number b is given by
the ABSOLUTE VALUE |b – a|; the length of a line segment
connecting points A = (a 1 ,a 2 ) and B = (b 1 ,b 2 ) is given
by the DISTANCE FORMULA
; and
the length of a VECTOR a = is given by ʈaʈ =
. (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 explaining 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 BLAISE PASCAL (1623–62) and
later contributed to the solution of the famous
BRACHISTOCHRONE problem. Before pursuing mathematics 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 BERNOULLI FAMILY to teach
him the newly discovered theory of calculus. This was
conducted chiefly by correspondence, and the agreement was made that all content of the letters sent
between them would belong to the marquis. This material 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 principles 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 geography as examples. Later chapters deal with cusps, points
of inflection, higher-order derivatives, evolutes, and
caustics. L’HÔPITAL’S RULE appears in chapter nine.
After l’Hôpital’s death Johann Bernoulli complained 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 GOTTFRIED WILHELM LEIBNIZ, 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 second book Traité analytique des sections coniques
(Analytical treatise on conic sections), which was published posthumously in 1720. He had also planned to
write a third text, one on the topic of integral calculus, 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 GUILLAUME FRANÇOIS ANTOINE L’HÔPITAL (1661–1704), a
student of the mathematician Johann Bernoulli of the
BERNOULLI 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 → a is 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:
lim
x
x
x
→
−
−
1
2 1
2
2
√a 1
2 + a 2
2
√(b 1 – a 1 )
2 + (b 2 – a 2 )
2
310 L’Hôpital, Guillaume François Antoine, marquis de