Bernoulli Family

1
SOLO
Johann
1667-1748
Daniel
1700-1782
Jacob
1654-1705
Johann II
1710-1790
Nicolaus II
1695-1720
Jacob II
1759-1789
Johann III
1744-1807
BERNOULLI FAMILY
SOLO HERMELIN
Run This
http://www.solohermelin.com

3
Niklaus
(1623 – 1708)
Nikolaus I
(1687 – 1759)
Jacob I
(1655 – 1705)
Nikolaus
(1662 – 1716)
Johann
(1667 – 1748)
Hieronymus
(1669 – 1760)
Eduard
(1867 – 1927)
Nikolaus II
(1695 – 1726)
Daniel
(1700 – 1782)
Johann II
(1710 – 1790)
Franz
(1705 – 1777)
Eva
(1903 – 1995)
Carl
Albrecht
(1868 – 1937)
Carl
Christoph
(1861 – 1923)
Jacob II
(1759 – 1809
August
Leonhard
(1879 – 1939)
Elisabeth
(1873 – 1935)
Hans
(1876 – 1959)
Johann
Jacob
(1831 – 1913)
Carl
Gustav
(1834 – 1878)
Johann III
(1744 – 1807
Daniel II
(1751 – 1834
Nikolas III
(1754 – 1841
Nikolaus
(1793 – 1876)
Hieronymus
(1735 – 1786
Johann Jacob
(1769 – 1853
Johann Jacob
(1802 – 1892
Emanuel
(1776 – 1844)
Johannes
(1785 – 1869)
Christoph
(1782 – 1863)
Leonhard
(1786 – 1852)
Leonhard
(1791 – 1871)
August Christoph
(1839 – 1921)
Karl Johann
(1835 – 1906)
Carl Christoph
(1809 – 1884)
Eduard
(1819 – 1899)
Theodor
(1837 – 1909)
http://de.wikipedia.org/wiki/Bernoulli
SOLO

4
Bernoullianum Basel
http://de.wikipedia.org/wiki/Bernoullianum
SOLO

5
http://curvebank.calstatela.edu/lemniscate
The mathematical lineage in Basel
is amazing. James taught his
brother John. John taught L'Hôpital
the famous rule, but John's best
known student was another native of
Basel, Leonhard Euler. Daniel
Bernoulli would earn 10 prestigious
awards from the French Académie
Royale des Sciences, a record only
matched by Euler. His cousin,
Nicholas Bernoulli, was the first to
pose the famous St.Petersburg
paradox. The chair of mathematics
at the University was held by a
Bernoulli for over a hundred years
(1687 - 1790).
Run This
SOLO

6
Basel Bernoullianum from bus station
http://de.wikipedia.org/wiki/Bernoullianum
SOLO

7
Nicolaus Bernoulli (1623-1708), Jacob, Nicolaus and
Johann’s father, inherited the spice business in Basel that
had been set up by his own father, first in Amsterdam and
then in Basel. The family, of Belgium origin, were refugees
fleeing from persecution by the Spanish rulers of the
Netherlands. Philip, the King of Spain, had sent the Duke
of Alba to the Netherlands in 1567 with a large army to
punish those opposed to Spanish rule, to enforce adherence
to Roman Catholicism, and to re-establish Philip's
authority. Alba set up the Council of Troubles which was a
court that condemned over 12000 people but most, like the
Bernoulli family who were of the Protestant faith, fled the
country.
SOLO

8
Jacob Bernoulli was compelled to study philosophy and theology by his
parents, which he greatly resented, and he graduated from the University
of Basel with a master's degree in philosophy in 1671 and a licentiate in
theology in 1676. Jacob
1654-1705In 1676, after taking his theology degree, Bernoulli moved to Geneva
where he worked as a tutor. He then traveled to France spending two
years studying with the followers of Descartes who were led at this
time by Malebranche.
Jacob Bernoulli
René Descartes
1596-1650
Robert Boyle
1627 - 1691
Nicolas Malebranche
1638 - 1715
Johann van
Waveren Hudde
Robert Hooke
1635 - 1703
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Jacob.html
. In 1681 Bernoulli traveled to the Netherlands
where he met many mathematicians including Hudde. Continuing his
studies with the leading mathematicians and scientists of Europe he
went to England where, among others, he met Boyle and Hooke.
At this time he was deeply interested in astronomy and produced a
work giving an incorrect theory of comets. As a result of his travels,
Bernoulli began a correspondence with many mathematicians which
he carried on over many years.
SOLO

9
Jacob
1654-1705
Jacob Bernoulli
Jacob Bernoulli returned to Switzerland and taught mechanics at
the University in Basel from 1683, giving a series of important
lectures on the mechanics of solids and liquids. Since his degree was
in theology it would have been natural for him to turn to the
Church, but although he was offered an appointment in the Church
he turned it down.
Frans van Schooten
1615 - 1660
John Wallis
1616 - 1703
Isaac Barrow
1630 - 1677
René Descartes
1596-1650
Bernoulli's real love was for mathematics and theoretical physics and it was in these
topics that he taught and researched. During this period he studied the leading
mathematical works of his time including Descartes' Géométrie and van Schooten’s
additional material in the Latin edition. Jacob Bernoulli also studied the work of Wallis
“Arithmetica Infinitorum” and Barrow “Lectiones Geometricae” and through these he
became interested in infinitesimal geometry. Jacob began publishing in the journal Acta
Eruditorum which was established in Leipzig in 1682.
SOLO

10
Jacob
1654-1705
Jacob and Johann Bernoulli
One of the most significant events concerning the mathematical studies of
Jacob Bernoulli occurred when his younger brother, Johann Bernoulli, began
to work on mathematical topics. Johann was told by his father to study
medicine but while he was studying that topic he asked his brother Jacob to
teach him mathematics.
Gottfried Leibniz
1646 - 1716
Ehrenfried Walter
von Tschirnhaus
1651 - 1708
Johann
1667-1748
Jacob Bernoulli was appointed professor of mathematics in Basel in 1687 and
the two brothers began to study the calculus as presented by Leibniz in his
1684 paper on the differential calculus in Nova Methodus pro Maximis et
Minimis, itemque Tangentibus... published in Acta Eruditorum. They also
studied the publications of von Tschirnhaus. It must be understood that
Leibniz's publications on the calculus were very obscure to mathematicians of
that time and the Bernoullis were the first to try to understand and apply
Leibniz's theories.
SOLO

11
Jacob
1654-1705
Jacob and Johann Bernoulli
Johann
1667-1748
Although Jacob and Johann both worked on similar problems their
relationship was soon to change from one of collaborators to one of rivals.
Johann Bernoulli’s boasts were the first cause of Jacob's attacks on him and
Jacob wrote that Johann was his pupil whose only achievements were to
repeat what his teacher had taught him. Of course this was a grossly unfair
statement. Jacob continued to attack his brother in print in a disgraceful and
unnecessary fashion, particularly after 1697. However he did not reserve
public criticism for his brother. He was critical of the university authorities
at Basel and again he was very public in making critical statements that, as
one would expect, left him in a difficult situation at the university. Jacob
probably felt that Johann was the more powerful mathematician of the two
and, this hurt since Jacob's nature meant that he always had to feel that he
was winning praise from all sides. Hofmann writes:-
Sensitivity, irritability, a mutual passion for criticism, and an exaggerated need for
recognition alienated the brothers, of whom Jacob had the slower but deeper
intellect.
As suggested by this quote the brothers were equally at fault in their quarrel. Johann
would have liked the chair of mathematics at Basel which Jacob held and he certainly
resented having to move to Holland in 1695. This was another factor in the complete
breakdown of relations in 1697.
SOLO

12
Mathematical Works of Jacob Bernoulli
1. In 1682 Jacob Bernoulli did research work on theory of gravitation,
comets, etc.
2. Since 1683, he was a regular contributor to “Journal des Sçavans” and
“Acta Eruditorium”, where he published many new theorems in algebra..
3. He was between the first contributors to the theory of probability. His first paper in
probability was published in 1685. “Ars conjectandi”, Jacob Bernoulli’s book in
theory of probability was published in 1713, after his death.
4. In 1689, he published research papers in the theory of infinite series, where
he
showed that the series 1/12
+1/22
+1/32
+… converges.
5. In 1690, he introduced for the first time the term “integral” in a paper published
in “Acta eruditorum” (instead of “Calculus summatorium” used by Leibniz).
6. In 1692, he found out the evolutes of parabola and of the logarithmic spiral.
7. In 1696, Jacob invented the method of solving the differential equation of the
form dy/dx + P y = Q yn
(Bernoulli’s form), where P and Q are functions of x or
constants. He used this to solve geometrical and mechanical problems.
Jacob
1654-1705
SOLO

13
Mathematical Works of Jacob Bernoulli (continue)
8. He introduced the idea of polar coordinates in
analytical geometry and used it to find the
properties of spiral shaped curves.
9. He developed the binomial expansion of (1+1/n)n
to show that it
converges to a number between 2 and 3 (actual limit is
e = 2.73).
11. Jacob found the formulae for finding the radius of
curvature in both Cartesian and polar coordinates. .
10. Jacob Bernoulli studied the curve “Lemniscate of Bernoulli”
r2
=a2
cos 2q
Jacob
1654-1705
Jacob Bernoulli
SOLO

14Jacob Bernoulli’s “Ars Conjectandi”
James is best known for the work
Ars Conjectandi (The Art of
Conjecture), published eight years
after his death in1713 by his
nephew Nicholas. In this work, he
described the known results in
probability theory and in
enumeration, often proving
alternative proofs of known results.
This work also includes the
application of probability theory to
games of chance and his
introduction of the theorem known
as the law of large numbers. The
terms Bernoulli trial and Bernoulli
numbers result from this work.
Bernoulli crater, on the Moon, is
also named after him jointly with
his brother Johann.
SOLO

15
SOLO Review of Probability
Bernoulli Trials – The Binomial Distribution
( )
( )
( ) ( ) knkknk
pp
k
n
pp
knk
n
nkp
−−
−





=−
−
= 11
!!
!
,
Jacob
Bernoulli
1654-1705
( ) ( ) ( )
!
,1
!
;;
00 k
k
i
e
ipkP
k
i
ik
i
λγλ
λλ
λ
+
=== ∑∑
=
−
=
( ) pnxE =
Probability Density Functions
Cumulative Distribution Function
Mean Value
Variance( ) ( )ppnxVar −= 1
( ) ( )∫ −= −
x
a
dtttxa
0
1
exp,γγ is the incomplete gamma function
Moment Generating Function
( ) ( )[ ]n
pjp −+=Φ 1exp ωω Distribution
Examples

16
SOLO Review of Probability
Bernoulli Trials – The Binomial Distribution (continue – 1)
p – probability of success (r = 1) of a given discrete trial
q – probability of failure (r=0) of the given discrete trial
1=+ qp
n – number of independent trials
( )nkp , – probability of k successes in n independent trials (Bernoulli Trials)
( )
( )
( ) ( ) knkknk
pp
k
n
pp
knk
n
nkp
−−
−





=−
−
= 11
!!
!
,
Using the binomial theorem we obtain
( ) ( )∑=
−
−





==+
n
k
knkn
pp
k
n
qp
0
11
therefore the previous distribution is called binomial distribution.
Jacob
Bernoulli
1654-1705
Given a random event r = {0,1}
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 k
( )nkP ,
The probability of k successful trials from n independent
trials is given by
The number of k successful trials from n independent trials is given by
( )!!
!
knk
n
k
n
−
=





with probability ( ) knk
pp
−
−1
to permutations
and Combinations
Distribution
Examples

17
Jacob
1654-1705
The Bernoulli numbers are among the most interesting and important
number sequences in mathematics. They first appeared in the
posthumous work "Ars Conjectandi" (1713) by Jakob Bernoulli (1654-
1705) in connection with sums of powers of consecutive integers (see
Bernoulli (1713) or D.E. Smith (1959)). Bernoulli numbers are
particularly important in number theory, especially in connection with
Fermat's last theorem (see, e.g., Ribenboim (1979)). They also appear in
the calculus of finite differences (Nörlund (1924)), in combinatorics
(Comtet (1970, 1974)), and in other fields.
Definitions and main properties of Bernoulli numbers can be found in a great number
of articles and books listed in this bibliography. Good introductions are given, e.g., in
Ireland and Rosen (1982, 1990), Rademacher (1973), and Nörlund (1924). A handy
collection of formulas is in Abramowitz and Stegun (1964). Some older books are
entirely devoted to Bernoulli numbers; among them are Chistyakov (1895), Nielsen
(1923), and Saalschütz (1893). One should, however, be aware of possible differences
in notation and indexing, especially in older publications.
Bernoulli Numbers
The Bernoulli numbers Bn play an important role in several topics of
mathematics. These numbers can be defined by the power series
SOLO

18
SOLO Bernoulli Polynomials
Jacob
Bernoulli
1654-1705
( )
( )
( )
( )
( )
( )
( )
0
1
2
2
3 2
3
4 3 2
4
5 4 3
5
6 5 4 2
6
1
1
2
1
6
3 1
2 2
1
2
30
5 5 1
2 3 6
5 1 1
3
2 2 42
B x
B x x
B x x x
B x x x x
B x x x x
B x x x x x
B x x x x x
=
= −
= − +
= − +
= − + −
= − + −
= − + − +
When evaluated at zero, these definitions correspond to the Bernoulli numbers,

19
SOLO
In polar form:
Differential Geometry in the 3D Euclidean Space
Planar Curves
Lemniscate of Bernoulli
The lemniscate, also called the lemniscate of Bernoulli, is a polar curve whose most
common form is the locus of points the product of whose distances from two fixed points
(called the foci) a distance 2c away is the constant c2
. This gives the Cartesian equation:
( ) ( )222222
2 yxcyx −=+
http://mathworld.wolfram.com/Lemniscate.html
( )θ2cos2 22
cr =
Jakob Bernoulli published an article in Acta Eruditorum in 1694 in
which he called this curve the lemniscus (Latin for "a pendant
ribbon"). Bernoulli was not aware that the curve he was describing
was a special case of Cassini Ovals which had been described by
Cassini in 1680. The general properties of the lemniscate were
discovered by G. Fagnano in 1750 (MacTutor Archive)
Jacob Bernoulli
1654-1705
The most general form of the lemniscate is a toric section of a torus.
http://www-groups.dcs.st-and.ac.uk/~history/Curves/Lemniscate.html
( )[ ] ( )[ ] 42222
cycxycx =+++−

20
Lemniscate of James Bernoulli
There are several methods for drawing a
lemniscate. The easiest is illustrated above.
Draw a circle and then extend a diameter to
become a secant. The center of the lemniscate
O will be √2 times the radius of the circle.
Through O draw several segments cutting the
circle. The pattern of the lemniscate emerges in
the first quadrant.
For the more mechanically
minded, we suggest using the
method described in E. H.
Lockwoods' book.
http://curvebank.calstatela.edu/lemniscate/lemniscate.htm Run This
SOLO

21
SOLO
Planar Curves
Equiangular Spiral
Equiangular spiral (also known as logarithmic spiral,
Bernoulli spiral, and logistique) describe a family of
spirals. It is defined as a curve that cuts all radii vectors
at a constant angle.
The famous Equiangular Spiral was discovered by
Descartes, its properties of self-reproduction by
James (Jacob) Bernoulli (1654-1705) who requested that
the curve be engraved upon his tomb with the phrase
"Eadem mutata resurgo" ("I shall arise the same,
though changed.")
http://xahlee.org/SpecialPlaneCurves_dir/EquiangularSpiral_dir/equiangularSpiral.html
Logarithmic Spiral
θb
ear =
http://en.wikipedia.org/wiki/Logarithmic_spiral
Nautilus Shells
René Descartes
1596 - 1650
Jacob Bernoulli
1654-1705
http://mathworld.wolfram.com/LogarithmicSpiral.html

22Jacob Bernoulli’s Tomb
Bernoulli chose
a figure of a
logarithmic
spiral and the
motto "Eadem
mutata resurgo"
("Changed and
yet the same, I
rise again") for
his gravestone;
the spiral
executed by the
stonemasons
was, however, an
Archimedian
spiral.
SOLO

23
SOLO
Johann Bernoulli
Johann Bernoulli
1667-1748
Johann Bernoulli was the tenth child of Nicolaus and Margaretha Bernoulli. He
was the brother of Jacob Bernoulli but Johann was twelve years younger than his
brother Jacob which meant that Jacob was already a young man while Johann
was still a child. The two brothers were to have an important influence on each
others mathematical development and it was particularly true that in his early
years Johann must have been greatly influenced by seeing Jacob head towards a
mathematical career despite the objections of his parents.
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Johann.html
Nicolaus and Margaretha Bernoulli tried to set Johann on the road to a business career but,
despite his father's strong pushing, Johann seemed to be totally unsuited to a future in business.
Johann's father had intended him to take over the family spice business and in 1682, when he was
15 years old, Johann worked in the spice trade for a year but, not liking the work, he did not do well.
It was with great reluctance that Johann's father agreed in 1683 to Johann entering the University
of Basel. The subject that Johann Bernoulli was to study at university was medicine, a topic that
many members of the Bernoulli family ended up studying despite their liking for mathematics and
mathematical physics.
At Basel University Johann took courses in medicine but he studied mathematics with his
brother Jacob. Jacob was lecturing on experimental physics at the University of Basel when
Johann entered the university and it soon became clear that Johann's time was mostly devoted
to studying Leibniz's papers on the calculus with his brother Jacob. After two years of studying
together Johann became the equal of his brother in mathematical skill.

24
SOLO
Johann Bernoulli (continue – 1)
In 1691 Johann went to Geneva where he lectured on the differential calculus.
From Geneva, Johann made his way to Paris and there he met mathematicians
in Malebranche's circle, where the focus of French mathematics was at that
time. There Johann met de l’Hôpital and they engaged in deep mathematical
conversations. Contrary to what is commonly said these days, de l’Hôpital was a
fine mathematician, perhaps the best mathematician in Paris at that time,
although he was not quite in the same class as Johann Bernoulli.
Guillaume François Antoine
Marquis de L'Hôpital
1661 - 1704
De l’Hôpital was delighted to discover that Johann Bernoulli understood the
new calculus methods that Leibniz had just published and he asked Johann
to teach him these methods. This Johann agreed to do and the lessons were
taught both in Paris and also at de l’Hôpital 's country house at Oucques.
Bernoulli received generous payment from de l’Hôpital for these lessons,
and indeed they were worth a lot for few other people would have been able
to have given them.
After Bernoulli returned to Basel he still continued his calculus lessons by
correspondence, and this did not come cheap for de l’Hôpital who paid
Bernoulli half a professor's salary for the instruction. However it did assure
de l’Hôpital of a place in the history of mathematics since he published the
first calculus book “Analyse des infiniment petits pour l'intelligence des
lignes courbes” (1696) which was based on the lessons that Johann
Bernoulli sent to him.
Gottfried Wilhelm
von Leibniz
(1646-1716)
Nicolas Malebranche
1638 - 1715

25
SOLO
Guillaume François Antoine
Marquis de L'Hôpital
1661 - 1704
Johann Bernoulli
1667-1748
De l’Hôpital published the first calculus book “Analyse des infiniment petits pour
l'intelligence des lignes courbes” (1696) which was based on the lessons that
Johann Bernoulli sent to him.
The well known de l’Hôpital 's rule is contained in this calculus book
and it is therefore a result of Johann Bernoulli. In fact proof that the
work was due to Bernoulli was not obtained until 1922 when a copy of
Johann Bernoulli's course made by his nephew Nicolaus (I) Bernoulli
was found in Basel. Bernoulli's course is virtually identical with de
l’Hôpital 's book but it is worth pointing out that de l’Hôpital had
corrected a number of errors such as Bernoulli's mistaken belief that the
integral of 1/x is finite. After de l’Hôpital's death in 1704 Bernoulli
protested strongly that he was the author of de l’Hôpital's calculus book.
It appears that the handsome payment de l’Hôpital e made to Bernoulli
carried with it conditions which prevented him speaking out earlier.
However, few believed Johann Bernoulli until the proofs discovered in
1922.
( )
( )
( )
( )
( )
( )
0
0 0
0
lim 0
lim lim
lim 0
x x
x x x x
x x
d f x
f x
d xf x
If then
g x d g xg x
d x
→
→ →
→
 
 ÷= 
   = 
 =    ÷
 
De l’Hôpital’s rule

26
SOLO
Johann Bernoulli had already solved the problem of the catenary which had been
posed by his brother in 1691. He had solved this in the same year that his brother
posed the problem and it was his first important mathematical result produced
independently of his brother, although it used ideas that Jacob had given when he
posed the problem. At this stage Johann and Jacob were learning much from each
other in a reasonably friendly rivalry which, a few years later, would descend into
open hostility. For example they worked together on caustic curves during 1692-93
although they did not publish the work jointly. Even at this stage the rivalry was
too severe to allow joint publications and they would never publish joint work at
any time despite working on similar topics.
A catenary is the curve in which a heavy uniform chain hangs.
Its equation is:
( )1
cosh
2
x x
y x e e−
= = +
Johann Bernoulli
1667-1748

27
SOLO
Johann Bernoulli
1667-1748
A stream of mathematical ideas continued to flow from Johann Bernoulli.
In 1694 he considered the function y = xx and he also investigated series using
the method of integration by parts. Integration to Bernoulli was simply viewed
as the inverse operation to differentiation and with this approach he had great
success in integrating differential equations. He summed series, and discovered
addition theorems for trigonometric and hyperbolic functions using the
differential equations they satisfy. This outstanding contribution to mathematics
reaped its reward in 1695 when he received two offers of chairs. He was offered a chair at Halle and
the chair of mathematics at Groningen. This latter chair was offered to Johann Bernoulli on the advice
of Huygens and it was this post which Johann accepted with great pleasure, not least because he now
had equal status to his brother Jacob who was rapidly becoming extremely jealous of Johann's
progress. The fault was not all on Jacob's side however, and Johann was equally to blame for the
deteriorating relations. It is interesting to note that Johann was appointed to the chair of mathematics
but his letter of appointment mentions his medical skills and offered him the chance to practice
medicine while in Groningen.
Johann Bernoulli had married Drothea Falkner and their first child was seven months old when the
family departed for Holland on 1 September 1695. This first child was Nicolaus (I) Bernoulli who also
went on to become a mathematician. Perhaps this is a good time to note that two other of Johann's
children went on to become mathematicians, Daniel Bernoulli, who was born while the family was in
Groningen, and Johann (II) Bernoulli.

28
HISTORY OF CALCULUS OF VARIATIONS
The brachistochrone problem
In 1696 proposed the Brachistochrone (“shortest time”)
Problem:
Given two points A and B in the vertical plane, what is the curve
traced by a point acted only by gravity, which starts at A and
reaches B in the shortest time.
Johann Bernoulli
1667-1748
SOLO
A
B
Run This

29
SOLO
H.J. Sussmann, J.C. Willems
“300 Years of Optimal Control:
From the Brachystochrone to the
Maximal Principle”,
IEEE Control Systems, June 1997,
pp. 32 - 44

30
Cycloid Equation
∫∫∫∫ 





=
+






+
===
cfcfcf xxxt
xd
xd
yd
yxFxd
ygV
xd
yd
V
sd
tdJ
00
2
0
2
00
,,
2
1
Minimization Problem
Solution of the Brachistochrone Problem:
SOLO

31
Jacob Bernoulli
(1654-1705)
Gottfried Wilhelm
von Leibniz
(1646-1716)
Isaac Newton
(1643-1727)
The solutions of Leibniz, Johann Bernoulli, Jacob Bernoulli
and Newton were published on May 1697 publication of
Acta Eruditorum. L’Hôpital solution was published only in 1988.
Guillaume François
Antoine de L’Hôpital
(1661-1704)
SOLO

32
SOLO
Johan Bernoulli (continue – 5)
Johann Bernoulli
1667-1748
In 1705 return to Basel Johann worked hard to ensure that he succeeded to his
brother's chair and soon he was appointed to Jacob's chair of mathematics. It is
worth remarking that Bernoulli's father-in-law lived for three years in which he
greatly enjoyed having his daughter and grandchildren back in Basel. There were
other offers that Johann turned down, such as Leiden, a second offer from Utrecht
and a generous offer for him to return to Groningen in 1717.
In 1713 Johann became involved in the Newton-Leibniz controversy. He strongly supported Leibniz
and added weight to the argument by showing the power of his calculus in solving certain problems
which Newton had failed to solve with his methods. Although Bernoulli was essentially correct in his
support of the superior calculus methods of Leibniz, he also supported Descartes' vortex theory over
Newton's theory of gravitation and here he was certainly incorrect. His support in fact delayed
acceptance of Newton's physics on the Continent.
Bernoulli also made important contributions to mechanics with his work on kinetic energy, which,
not surprisingly, was another topic on which mathematicians argued over for many years. His work
Hydraulica is another sign of his jealous nature. The work is dated 1732 but this is incorrect and
was an attempt by Johann to obtain priority over his own son Daniel. Daniel Bernoulli completed
his most important work Hydrodynamica in 1734 and published it in 1738 at about the same time as
Johann published Hydraulica. This was not an isolated incident, and as he had competed with his
brother, he now competed with his own son. As a study of the historical records has justified
Johann's claims to be the author of de l'Hôpital's calculus book, so it has shown that his claims to
have published Hydraulica before his son wrote Hydrodynamica are false.

33
http://curvebank.calstatela.edu/lemniscate
Johann BernoulliJohann Bernoulli
SOLO

34
SOLO
Planar Curves
Astroid
The Astroid was studied by Johan Bernoulli (1667 – 1748) ,by
D’Alembert in 1748. The name was given by Littrow in 1838.
Johann Bernoulli
1667-1748
Jean Le Rond D’Alembert
1717 - 1783



=
=
θ
θ
3
3
sin
cos
ay
ax
The Astroid can be obtained as a hypocycloid with b/a=1/4 or 3/4
The astroid is the curve performed by
a bus door
The astroid is the curve obtained by the
intersection of two circles rolling without
slipping inside a bigger circle. Run This

35
Daniel Bernoulli
Daniel Bernoulli
1700-1782
1700 - Born in Groningen, the son of Johann Bernoulli, nephew of
Jakob Bernoulli, younger brother of Nicolaus II Bernoulli,
and older brother of Johann II,
1713 - Daniel was sent to Basel University at the age of 13 to study
philosophy and logic.
1715 - He obtained his baccalaureate examinations.
1716 - He obtained his master's degree . During the time he studied
philosophy at Basel, he was learning the methods of the calculus
from his father and his older brother Nicolaus(II) Bernoulli.
1718/9 - studied medicine at Heidelberg and Strasbourg
1720 - He returned to Basel to complete his doctorate in medicine. By this stage Johann
Bernoulli was prepared to teach his son more mathematics while he studied medicine
and Daniel studied his father's theories of kinetic energy. What he learned on the
conservation of energy from his father he applied to his medical studies and Daniel
wrote his doctoral dissertation on the mechanics of breathing. So like his father Daniel
had applied mathematical physics to medicine in order to obtain his medical doctorate.
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Daniel.html
1720-25 Venice Italy where he worked on mathematics
SOLO

36
Daniel Bernoulli
Daniel Bernoulli
1700-1782
1724 - Mathematical Exercises was published, with Goldbach's assistance,.
This consisted of four separate parts
The first part described the game of faro and is of little
importance other than showing that Daniel was learning about
probability at this time. The second part was on the flow of water
from a hole in a container and discussed Newton's theories
(which were incorrect). Daniel had not solved the problem of
pressure by this time but again the work shows that his interest
was moving in this direction. His medical work on the flow of
blood and blood pressure also gave him an interest in fluid flow.
The third part of Mathematical exercises was on the Riccati differential equation while the
final part was on a geometry question concerning figures bounded by two arcs of a circle.
1725 Return to Basel from Venice. In Venice, Daniel had also designed
an hour glass to be used at sea so that the trickle of sand was
constant even when the ship was rolling in heavy seas
He submitted his work on this to the Paris Academy and in 1725
won the first prize.
Daniel had also attained fame through his work Mathematical
exercises and on the strength of this he was invited to take up the
chair of mathematics at St Petersburg.
His brother Nicolaus(II) Bernoulli was also offered a chair of
mathematics at St Petersburg so in late 1725 the two brothers
traveled to St Petersburg. Within eight months of their taking up
the appointments in St Petersburg Daniel's brother died of fever.
Nicolaus II
1695-1720
SOLO

37
Daniel Bernoulli
Daniel Bernoulli
1700-1782
1725 - 33 Daniel Bernoulli stayed at St Petersburg.
Leonhard Euler
1707 - 1783
1727 - Leonard Euler a pupil of his father Johann Bernoulli joined him at
St Petersburg.
1728 - Bernoulli and Euler dominated the mechanics of flexible and
elastic bodies, in that year deriving the equilibrium curves for
these bodies
0
0
0 & 0x
x
u
u
x=
=
∂
= =
∂
)fixed end(
2 3
2 3
0 & 0
x L x L
u u
x x= =
∂ ∂
= =
∂ ∂
)free end(
2
2
u
EI F
x x
 ∂ ∂
− = ÷
∂ ∂ 
http://en.wikipedia.org/wiki/Euler-Bernoulli_beam_equation
Euler – Bernoulli beam
While in St Petersburg he made one of his most famous discoveries
when he defined the simple nodes and the frequencies of oscillation
of a system. He showed that the movements of strings of musical
instruments are composed of an infinite number of harmonic
vibrations all superimposed on the string.
SOLO

38
Daniel Bernoulli
Daniel Bernoulli
1700-1782
1725 - 33 Daniel Bernoulli stayed at St Petersburg.
A second important work which Daniel produced while in St
Petersburg was one on probability and political economy. Daniel
makes the assumption that the moral value of the increase in a
person's wealth is inversely proportional to the amount of that
wealth. He then assigns probabilities to the various means that a
person has to make money and deduces an expectation of increase in
moral expectation. Daniel applied some of his deductions to
insurance.
Undoubtedly the most important work which Daniel Bernoulli did while in St Petersburg was his
work on hydrodynamics. Even the term itself is based on the title of the work which he produced
called Hydrodynamica and, before he left St Petersburg, Daniel left a draft copy of the book with a
printer. However the work was not published until 1738 and although he revised it considerably
between 1734 and 1738, it is more the presentation that he changed rather then the substance.
2
2
v
g h p const
ρ
ρ+ + =
Bernoulli’s Principle
http://en.wikipedia.org/wiki/Bernoulli%27s_principle
SOLO

39
Bernoulli's diagram to illustrate how pressure
is measured.
See also part of
Bernoulli's original Latin explanation.
Source: The Turner Collection, Keele
University
Daniel Bernoulli (1700-1782)
http://plus.maths.org/issue1/bern/
His chief work is his Hydrodynamique (Hydrodynamica), published in 1738;
it resembles Joseph Louis Lagrange's Méchanique Analytique in being
arranged so that all the results are consequences of a single principle,
namely, conservation of energy. This was followed by a memoir on the
theory of the tides, to which, conjointly with the memoirs by Euler and Colin
Maclaurin, a prize was awarded by the French Academy: these three
memoirs contain all that was done on this subject between the publication of
Isaac Newton's Philosophiae Naturalis Principia Mathematica and the
investigations of Pierre-Simon Laplace.
Daniel Bernoulli (continue – 2) SOLO

40
KINETIC THEORY OF GASES
HISTORY
STATE EQUATION )BOYLE-MARIOTTE LAW(
p - PRESSURE (FORCE / SURFACE)
V - VOLUME OF GAS
m - MASS OF GAS
T - GAS TEMPERATURE
-GAS DENSITY
[ ]m3
[ ]kg
[ ]o
K
[ ]kg m/ 3
ρ
[ ]2
/ mN
ROBERT BOYLE )1660( DISCOVERED EXPERIMENTALLY THAT THE
PRODUCT OF PRESSURE AND VOLUME IS CONSTANT FOR A FIXED
MASS OF GAS AT CONSTANT TEMPERATURE
SOLO
( )mTconstVp ,=
ROBERT BOYLE )1627-1691(
New Experiments Physio-Mechanicall,
Touching the Spring of the Air, and
Its Effects )1660(

41
KINETIC THEORY OF GASES
HISTORY
RICHARD TOWNLEY )1628-1707( AND HENRY POWER )1623-1668( FOUND
ALSO EXPERIMENTALLY THE P*V LAW IN 1660.
SOLO
STATE EQUATION )BOYLE-MARIOTTE LAW(
EDMÉ MARIOTTE )~1620-1684( INDEPENDENTLY FINDS THE
RELATIONSHIP BETWEEN PRESSURE AND VOLUME P*V LAW IN 1676
IN HIS WORK “On the Nature of Air”.
( )mTconstVp ,=
THIS IS KNOWN AS “MARIOTTE’s LAW” IN FRANCE AND “BOYLE’s LAW”
ELSEWHERE.
DANIEL BERNOULLI )1700-1782( IN THE TREATISE
“Hydrodynamica” )1738( DERIVES THE
BOYLE-MARIOTTE LAW USING A “BILLIARD BALL”
MODEL. HE ALSO USES CONSERVATION OF
MECHANICAL ENERGY TO SHOW THAT THE PRESSURE
CHANGES PROPORTIONALLY TO THE SQUARE OF
PARTICLE VELOCITIES AS TEMPERATURE CHANGES.

42
Daniel Bernoulli
Daniel Bernoulli
1700-1782
1733 - He left St Petersburg with his younger brother Johann(II) Bernoulli
who was also with him in St Petersburg and they, making visits to
Danzig, Hamburg, Holland and Paris before returning to Basel in 1734
Johann II
1710-1790
1734 - Daniel Bernoulli submitted an entry for the Grand Prize of the
Paris Academy for 1734 giving an application of his ideas to
astronomy. This had unfortunate consequences since Daniel's
father, Johann Bernoulli, also entered for the prize and their
entries were declared joint winners of the Grand Prize. The result
of this episode of the prize of the Paris Academy had unhappy
consequences for Daniel. His father was furious to think that his
son had been rated as his equal and this resulted in a breakdown
in relationships between the two. The outcome was that Daniel
found himself back in Basel but banned from his father's house.
Whether this caused Daniel to become less interested in
mathematics or whether it was the fact that his academic position
was a non mathematical one, certainly Daniel never regained the
vigour for mathematical research that he showed in St Petersburg.
SOLO

43
Daniel Bernoulli
Daniel Bernoulli
1700-1782
1733 - Although Daniel had left St Petersburg, he began an immediate
correspondence with Euler and the two exchanged many ideas on
vibrating systems. Euler used his great analytic skills to put many
of Daniel's physical insights into a rigorous mathematical form.
Daniel continued to work on polishing his masterpiece
Hydrodynamica for publication and added a chapter on the force
of reaction of a jet of fluid and the force of a jet of water on an
inclined plane. In this chapter, Chapter 13, he also discussed
applications to the propulsion of ships.
1737 - The prize of the Paris Academy also had a nautical theme,
the best shape for a ship's anchor, and Daniel Bernoulli was again
the joint winner of this prize, this time jointly with Poleni
Botany lectures were not what Daniel wanted and things became better for him in 1743 when he was
able to exchange these for physiology lectures. In 1750, however, he was appointed to the chair of
physics and taught physics at Basel for 26 years until 1776. He gave some remarkable physics lectures
with experiments performed during the lectures. Based on experimental evidence he was able to
conjecture certain laws which were not verified until many years later. Among these was Coulomb's law
in electrostatics.
SOLO

44
Daniel Bernoulli
Daniel Bernoulli
1700-1782
Daniel Bernoulli did produce other excellent scientific work during these
many years back in Basel. In total he won the Grand Prize of the Paris
Academy 10 times, for topics in astronomy and nautical topics.
1740 - won (jointly with Euler) the Grand Prize of the Paris Academy for
work on Newton's theory of the tides
1743 - won the Grand Prize of the Paris Academy for essays on magnetism
1746 - won the Grand Prize of the Paris Academy for essays on magnetism
1747 - won the Grand Prize of the Paris Academy for a method to determine time at sea
1751 - won the Grand Prize of the Paris Academy for an essay on ocean currents
1753 - won the Grand Prize of the Paris Academy for the effects of forces on ships
1757 - won the Grand Prize of the Paris Academy for proposals to reduce the pitching and tossing
of a ship in high seas.
SOLO

45
Daniel Bernoulli
Daniel Bernoulli
1700-1782
Another important aspect of Daniel Bernoulli's work that proved
important in the development of mathematical physics was his acceptance
of many of Newton's theories and his use of these together with the tolls
coming from the more powerful calculus of Leibniz. Daniel worked on
mechanics and again used the principle of conservation of energy which
gave an integral of Newton's basic equations. He also studied the
movement of bodies in a resisting medium using Newton's methods.
Isaac Newton
1643-1727
He also continued to produce good work on the theory of oscillations
and in a paper he gave a beautiful account of the oscillation of air in
organ pipes
Daniel Bernoulli was much honoured in his own lifetime. He was elected
to most of the leading scientific societies of his day including those in
Bologna, St Petersburg, Berlin, Paris, London, Bern, Turin, Zurich and
Mannheim.
SOLO

47
Nicolaus II Bernoulli
Nicolaus, who showed great promise in the field of mathematics, was called to the St.Petersburg
Academy, in 1727, where he unfortunately died of fever, only eight months later. He wrote on
curves, differential equations, and probability.
Nicolaus(II) Bernoulli was the favourite of three sons of Johann Bernoulli.
He entered the University of Basel when only 13 years of age and, like
many other members of his family, studied both mathematics and law. In
1715 he became a licentiate in jurisprudence.
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Nicolaus(II).html
Nicolaus worked as his father's assistant helping him with correspondence.
In particular he was involved with writing letters concerning the famous
priority dispute between Newton and Leibniz. He not only replied to Taylor
regarding the dispute but he also made important mathematical contributions to the problem of
trajectories while working on the mathematical arguments behind the dispute.
Nicolaus worked on curves, differential equations and probability. He died only 8 months after
taking up an appointment in St Petersburg at a young age when his talents promised so much
for the future.
Nicolaus II
1695-1720
SOLO

48
Johann II Bernoulli
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Johann(II).html
http://library.thinkquest.org/22584/temh3007.htm
Johann II, the youngest of the three sons, of Johann Bernoulli,
studied law but spent his later years as a professor of mathematics at
the University of Basel. He was particularly interested in the
mathematical theory of heat and light.
Johann II
1710-1790
He worked on mathematics both with his father and as an independent
worker. He had the remarkable distinction of winning the Prize of the Paris
Academy on no less than four separate occasions. On the strength of this he
was appointed to his father's chair in Basel when Johann Bernoulli died.
SOLO

49
Nicolaus I Bernoulli
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Nicolaus(I).html
Nicolaus(I) Bernoulli (1687-1759), was a nephew of Jacob Bernoulli and Johann Bernoulli.
His early education involved studying mathematics with his uncles. In fact it was Jacob Bernoulli
who supervised Nicolaus's Master's degree at the University of Basel which he was awarded in
1704. Five years later he was received a doctorate for a dissertation which studied the application
of probability theory to certain legal questions.
Nicolaus Bernoulli was appointed to Galileo's chair at Padua in 1716 which Hermann had filled
immediately prior to Nicolaus's appointment. There he worked on geometry and differential
equations. In 1722 he left Italy and returned to his home town to take up the chair of logic at the
University of Basel. After nine years, remaining at the University of Basel, he was appointed to
the chair of law. In addition to these academic appointments, he did four periods as rector of the
university.
From Montmort's work we can see that Nicolaus formulated certain problems in the theory of
probability, in particular the problem which today is known as the St Petersburg problem. Nicolaus
also corresponded with Leibniz during the years 1712 to 1716. In these letters Nicolaus discussed
questions of convergence, and showed that (1+x)n diverges for x > 0.
A problem in probability, which he proposed from St. Peresburg, later became known as the
Petersburg paradox. The problem is:If A receives a penny when a head appears on the first toss of
a coin, two pennies if a head does not appear until second toss, four pennies if a head does not
appear until the third toss, and so on, what is A's expectation? Mathematical theory shows that
A's expectation is infinite, which seems a paradoxical result. The problem was incestigated by
Nicolaus' brother Daniel, who succeeded Nicolaus at St. Petersburg.
http://library.thinkquest.org/22584/temh3007.htm http://en.wikipedia.org/wiki/St._Petersburg_Paradox
SOLO

50
Nicolaus I Bernoulli
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Nicolaus(I).html
In his letters to Euler (1742-43) he criticises Euler's indiscriminate use of divergent series. In this
correspondence he also solved the problem of the sum of the reciprocal squares ∑ (1/n2) = π2/6,
which had confounded Leibniz and Jacob Bernoulli.
Nicolaus Bernoulli assisted in the publication of Jacob Bernoulli's Ars conjectandi. Later
Nicolaus edited Jacob Bernoulli's complete works and supplemented it with results taken from
Jacob's diary. Other problems he worked on involved differential equations. He studied the
problem of orthogonal trajectories, making important contributions by the construction of
orthogonal trajectories to families of curves, and he proved the equality of mixed second-order
partial derivatives. He also made significant contributions in studying the Riccati equation.
One of the great controversies of the time was the Newton Leibniz argument. As might be expected
Nicolaus supported Leibniz but he did produce some good arguments in his favour such as
observing that Newton failed to understand higher derivatives properly which had led him into
errors in the problem of inverse central force in a resisting medium.
Nicolaus(I) Bernoulli received many honours for his work. For example he was elected a member
of the Berlin Academy in 1713, a Fellow of the Royal Society of London in 1714, and a member of
the Academy of Bologna in 1724.
SOLO

51
Johann III Bernoulli
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Johann(III).html
Johann Bernoulli II had a son Johann III (1744-1807) who, like his
father, studied law but then turned to mathematics. When barely
nineteen years old, he was called as a professor of mathematics to the
Berlin Academy. He wrote on astonomy, the doctrine of chance,
recurring decimals, and indeterminate equations..
Johann III
1744-1807
In the field of mathematics he worked on probability, recurring decimals and the theory of
equations. As in his astronomical work there was little of lasting importance. He did, however,
publish the Leipzig Journal for Pure and Applied Mathematics between 1776 and 1789.
He was well aware of the famous mathematical line from which he was descended and he
looked after the wealth of mathematical writings that had passed between members of the
family. He sold the letters to the Stockholm Academy where they remained forgotten about until
1877. At that time when these treasures were examined, 2800 letters written by Johann(III)
Bernoulli himself were found in the collection.
SOLO

52
Jacob II Bernoulli
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Jacob(II).html
Jacob (II) Bernoulli was one of the sons of Johann(II) Bernoulli.
Following the family tradition he took a degree in law but his interests
were in mathematics and mathematical physics.
In 1782 Jacob(II) Bernoulli's uncle Daniel Bernoulli died and his
chair of physics in Basel became vacant. Jacob(II) applied for the
chair and presented a work on mathematical physics to support his
application. The decison as to who should fill the vacant chair was not
made on academic grounds but was made by drawing lots. Jacob(II)
Bernoulli was unlucky and he was not offered this position he would
really have liked.
He was then appointed as secretary to the Imperial Envoy to Turin and
Venice. However, he was soon given the chance of another academic post
when he received an offer from St Petersburg. He went to St Petersburg and
began to write important works on mathematical physics which he presented
to the St Petersburg Academy of Sciences. These treatises were on elasticity,
hydrostatics and ballistics.
Despite the rather harsh climate, the city of St Petersburg had great attractions for Jacob(II)
Bernoulli since his uncle Daniel Bernoulli had worked there with Euler. In fact Jacob(II) married a
granddaughter of Euler in St Petersburg but, tragically, the city was to lead to his death.
Jacob II
1759-1789
http://www.ub.unibas.ch/spez/bernoull.htm
SOLO

53
Lesser Bernoulli descendants are Daniel II (1751-1834) and Jakob II
(1759-1789), two other sons of Johann II, Christoph (1782-1834), a son
of Daniel II, and Johann Gustav (1811-1863), a son of Christoph.
SOLO

54
Burial-place of Johann III Bernoulli in Berlin-Köpenick
http://www.w-volk.de/museum/grave05.htm
Johann III Bernoulli was buried on the cemetery of the St. Laurentius Church in Berlin-Köpenick. It seems that the grave is not
longer existant but the metallic cross with the names and the dates of him and his wife is still there.
SOLO

55
http://www.math.wichita.edu/history/men/bernoulli.html
http://www.maths.tcd.ie/pub/HistMath/People/Bernoullis/RouseBall/RB_Bernoullis.html
http://library.thinkquest.org/27694/Bernoulli%20Family.htm
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Daniel.html
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Johann(II).html
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Nicolaus(I).html
http://www.bookrags.com/biography/jakob-bernoulli/
http://www.fluidmech.net/tutorials/bernoulli/bernoulli-history.htm
http://www.roma.unisa.edu.au/07305/fhome.htm
http://en.wikipedia.org/wiki/Bernoulli%27s_principle
http://en.wikipedia.org/wiki/Euler-Bernoulli_beam_equation
http://en.wikipedia.org/wiki/Daniel_Bernoulli
http://www.fyma.ucl.ac.be/~gaino/Bernoulli/aboutus.html
SOLO

January 6, 2015 56
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA

Bernoulli Family

More Related Content

Viewers also liked

Similar to Bernoulli Family

More from Solo Hermelin

Recently uploaded

Bernoulli Family

Editor's Notes