posthumously in 1713. The B
ERNOULLI NUMBERS
also
appear, for the first time, in this text. Jacob occupied
the chair of mathematics at Basel University from
1687 until his death.
Johann (July 17, 1667–January 1, 1748) is also
known for his work on calculus. Being recognized as
an expert in the field, Johann was hired by the French
nobleman M
ARQUIS DE
G
UILLAUME
F
RANÇOIS
A
NTOINE L
’H
ÔPITAL
(1661–1704) to explain the new
theory to him, first through formal tutoring sessions in
Paris, and then through correspondence when Johann
later returned to Basel. L’Hôpital published the con-
tents of the letters in a 1696 textbook Analyse des
infiniment petits (Analysis with infinitely small quanti-
ties), but gave little acknowledgment to Johann. The
famous rule that now bears his name,
L
’H
ÔPITAL
’
S
RULE
, is due to Johann. Johann succeeded his brother
in the chair at Basel University.
The two brothers, Jacob and Johann, worked on
similar problems, and each maintained an almost
constant exchange of ideas with Leibniz. The rela-
tionship between the two siblings, however, was not
amicable, and they often publicly criticized each
other’s work.
Nicolaus (I) Bernoulli (October 21, 1687–November
29, 1759), nephew to Jacob and Johann, also achieved
some fame in mathematics. He worked on problems in
GEOMETRY
,
DIFFERENTIAL EQUATION
s, infinite
SERIES
,
and probability. He held the chair of mathematics at
Padua University, once filled by G
ALILEO
G
ALILEI
(1564–1642).
Johann Bernoulli had three sons, all of whom
themselves became prominent mathematicians:
Nicolaus (II) Bernoulli (February 6, 1695–July 31,
1726) wrote on curves, differential equations, and
probability theory. He died—by drowning while swim-
ming—only eight months after accepting a prestigious
appointment at the St. Petersberg Academy.
Daniel Bernoulli (February 8, 1700–March 17,
1782), the most famous of the three sons, is noted for
his 1738 text Hydrodynamica (Hydrodynamics),
which laid the foundations for the modern discipline
of hydrodynamics. (Daniel’s father, Johann, jealous of
his son’s success, published his own text on hydrody-
namics in 1739 but placed on it the publishing date
of 1732 and accused his son of plagiarism.) Daniel
also worked on the mathematics of vibrating strings,
the kinetic theory of gases, probability theory, and
partial differential equations. He was awarded the
Grand Prize from the Paris Academy no fewer than
10 times.
Johann (II) Bernoulli (May 28, 1710–July 17,
1790) studied the mathematics of heat flow and light.
He was awarded the Grand Prize from the Paris
Academy four times and succeeded his father in the
chair at Basel University in 1743.
Johann (II) Bernoulli had three sons, two of whom,
Johann (III) Bernoulli (November 4, 1744–July 13,
1807) and Jacob (II) Bernoulli (October 17, 1759–
August 15, 1789), worked in mathematics. Johann (III)
studied astronomy and probability, and wrote on recur-
ring decimals and the theory of equations. He was a
professor of mathematics at Berlin University at the
young age of 19. Jacob (II) Bernoulli wrote works on
the mathematics of elasticity, hydrostatics, and ballis-
tics. He was professor of mathematics at the St. Peters-
burg Academy, but, like his uncle, drowned at the age
of 29 while swimming in the Neva River.
Members of the Bernoulli family had a profound
effect on the early development of probability theory,
calculus, and the field of continuum mechanics. Many
concepts (such as the Bernoulli numbers, a probability
distribution, a particular differential equation) are
named in their honor.
Bernoulli numbers See
SUMS OF POWERS
.
Bertrand’s paradox French mathematician Joseph-
Louis François Bertrand (1822–1900) posed the fol-
lowing challenge:
40 Bernoulli numbers
Answering Bertrand’s paradox