58 cancellation
he had developed these ideas as early as 1665, but he did
not publish an account of his theory until 1704. Unfortunately,
his writing style and choice of notation also made his ver-
sion of calculus accessible only to a select audience. Leib-
niz, on the other hand, made explicit use of an infinitesimal in
his development of the theory. He called the infinitesimal
change of a quantity xa
DIFFERENTIAL
, denoted dx. Leibniz
invented a beautiful notational system for the subject that
made reading and working with his account of the theory
immediately accessible to a wide audience. (Many of the
symbols we use today in differential and integral calculus
are due to Leibniz.) Leibniz formulated his approach in the
mid-1670s and published his account of the subject in 1684.
Although it is now known that Newton and Leibniz had made
their discoveries independently, matters at the time were not
clear, and a bitter dispute arose over the priority for the dis-
covery of calculus. In 1712 the R
OYAL
S
OCIETY
of England
formed a special committee to adjudicate the issue.
Applying the techniques to problems of the real world
became the main theme of 18th-century mathematics. New-
ton’s famous 1687 text Principia paved the way with its anal-
ysis of the laws of motion and the mechanics of the solar
system. The Swiss brothers Jakob Bernoulli (1654–1705)
and Johann Bernoulli (1667–1748) of the famous B
ERNOULLI
FAMILY
, champions of Leibniz in the famous dispute, studied
the newly invented calculus and were the first to give public
lectures on the topic. Johann Bernoulli was hired to teach
differential calculus to the French nobleman G
UILLAUME
F
RANÇOIS DE L
’H
OPITAL
(1661–1704) via written correspon-
dence. In 1696 L’Hopital then published the content of
Johann’s letters with his own name as author. Italian math-
ematician M
ARIE
G
AETANA
A
GNESI
(1718–99) wrote the first
comprehensive textbook dealing with both differential and
integral calculus in 1755.
The Swiss mathematician L
EONHARD
E
ULER
(1707–83) and
French mathematicians J
OSEPH
-L
OUIS
L
AGRANGE
(1736–1813)
and P
IERRE
-S
IMON
L
APLACE
(1749–1827) were prominent in
developing the theory of
DIFFERENTIAL EQUATION
s. Euler also
wrote extensively on the subject of calculus, showing how
the theory can be applied to a vast range of pure and applied
mathematical problems. Yet despite the evident success of
calculus, some 18th-century scholars questioned the validity
and the soundness of the subject.
The sharpest critic of Newton’s and Leibniz’s work was
the Anglican Bishop of Coyne, George Berkeley (1685–1753).
In his scathing essay, “The Analyst,” Berkeley demon-
strated, convincingly, that both Newton’s notion of a fluxion
and Leibniz’s concept of an infinitesimal are ill-defined, and
that the foundations of the subject are consequently inse-
cure. (Some historians suggest that Berkeley’s vehement
criticisms were motivated by a personal disdain for the
apparent atheism of the type of mathematician who argues
that science is certain and that theology is based on specu-
lation.) Mathematicians consequently began looking for
ways to put calculus on a sound footing. Significant
progress was not made until the 19th century, when French
mathematician A
UGUSTINE
L
OUIS
C
AUCHY
(1789–1857) sug-
gested that the notion of an infinitesimal should be replaced
by that of a
LIMIT
. German mathematician K
ARL
W
EIERSTRASS
(1815–97) developed this idea further and was the first to
give absolutely clear and precise definitions to all concepts
used in calculus, devoid of any mystery or reliance on geo-
metric intuition. The work of German mathematician R
ICHARD
D
EDEKIND
(1831–1916) highlighted the role properties of the
real number system play in ensuring the validity of the
INTER
-
MEDIATE
-
VALUE THEOREM
and
EXTREME
-
VALUE THEOREM
and all
the essential results that follow from them.
Initially, calculus was deemed a theory pertaining only
to continuous change and
CONTINUOUS FUNCTION
s. German
mathematician B
ERNHARD
R
IEMANN
(1826–66) was the first to
consider, and give careful discussion on, the integration of
discontinuous functions. His definition of an integral is the
one typically presented in textbooks today. At the end of the
19th century, French mathematician H
ENRI
L
ÉON
L
EBESGUE
(1875–1941) literally turned Riemann’s approach around and
developed a concept of integration that can be applied to a
much wider class of functions and class of settings. In con-
structing a Riemann integral, one begins by subdividing the
range of inputs, the x-axis, into small intervals and adding
areas of rectangles above these intervals of heights given
by the function. This is akin to counting the value of a pock-
etful of coins by taking one coin out at a time, and adding
the outcomes as one goes along. Lebesgue suggested, on
the other hand, subdividing the range of outputs, the y-axis,
into small intervals and measuring the size of the sets on
the x-axis for which the function gives the desired output on
the y-axis. This is akin to counting coins by first collecting
all the pennies and determining their number, all the nickels
and ascertaining the size of that collection, and so forth. In
order to do this, Lebesgue had to develop a general “mea-
sure theory” for determining the size of complicated sets.
His new theory proved to be fundamentally important, and it
now has profound applications to a wide range of mathe-
matical topics. It proved to be especially important to the
sound development of
PROBABILITY
theory.
See also
CALCULUS
;
DIFFERENTIAL CALCULUS
;
GRAPH OF A
FUNCTION
;
INTEGRAL CALCULUS
;
VOLUME
.
History of Calculus
(continued)