Bernoulli Family

1

SOLO

Johann1667-1748

Daniel1700-1782

Jacob1654-1705

Johann II1710-1790

Nicolaus II1695-1720

Jacob II1759-1789

Johann III1744-1807

BERNOULLI FAMILYSOLO HERMELIN

Run This

2

BERNOULLI FAMILYSOLO

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)

CarlAlbrecht

(1868 – 1937)

CarlChristoph

(1861 – 1923)

Jacob II(1759 – 1809

AugustLeonhard

(1879 – 1939)

Elisabeth(1873 – 1935)

Hans(1876 – 1959)

JohannJacob

(1831 – 1913)

CarlGustav

(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

http://upload.wikimedia.org/wikipedia/de/c/ce/Bernoullianum_Basel.jpg

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 Boyle1627 - 1691

Nicolas Malebranche1638 - 1715

Johann vanWaveren Hudde

1628 - 1724

Robert Hooke1635 - 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

Jacob1654-1705

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 Schooten1615 - 1660

John Wallis1616 - 1703

Isaac Barrow1630 - 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

Jacob1654-1705

Jacob and Johann Bernoullihttp://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Jacob.html

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 Leibniz1646 - 1716

Ehrenfried Walter von Tschirnhaus

1651 - 1708

Johann1667-1748

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

Jacob1654-1705

Jacob and Johann Bernoulli


Johann1667-1748

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.

Jacob1654-1705

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

Jacob1654-1705

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 ppk

npp

knk

nnkp

11

!!

!,

JacobBernoulli1654-1705

!

,1

!;;

00 k

k

i

eipkP

k

i

ik

i

pnxE

Probability Density Functions

Cumulative Distribution Function

Mean Value

Variance ppnxVar 1

x

a dtttxa0

1 exp,γ is the incomplete gamma function

Moment Generating Function

npjp 1exp DistributionExamples

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 ppk

npp

knk

nnkp

11

!!

!,

Using the binomial theorem we obtain

n

k

knkn ppk

nqp

0

11

therefore the previous distribution is called binomial distribution.


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

DistributionExamples

17

Jacob1654-1705

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


0

1

22

3 23

4 3 24

5 4 35

6 5 4 26

1

1

21

63 1

2 21

230

5 5 1

2 3 65 1 1

32 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

2cos2 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 Bernoulli1654-1705

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

SOLODifferential Geometry in the 3D Euclidean Space

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

bear http://en.wikipedia.org/wiki/Logarithmic_spiral

Nautilus Shells

René Descartes1596 - 1650

Jacob Bernoulli1654-1705

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 Bernoulli1667-1748

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

SOLOJohann 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 Wilhelmvon Leibniz(1646-1716)

Nicolas Malebranche1638 - 1715

25

SOLO

Guillaume François Antoine Marquis de L'Hôpital

1661 - 1704

Johann Bernoulli (continue – 2)



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 0lim lim

lim 0

x x

x x x x

x x

d f xf x d xf x

If theng x d g xg x

d x

De l’Hôpital’s rule

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/De_L'Hopital.html

26



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

cosh2

x xy x e e


http://upload.wikimedia.org/wikipedia/commons/1/1d/Johann_Bernoulli.jpg

27




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

x

y

V

t

fcfc yxB ,fcx

fcy

N

0,0A

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 theMaximal Principle”,IEEE Control Systems, June 1997,pp. 32 - 44

30

HISTORY OF CALCULUS OF VARIATIONS

The brachistochrone problem

sinr

cosr

r

x

y

0x

0V

g

V

2

20

r

rA

B

),( yx

g

Vry

rxx

2cos1

sin2

0

0

Cycloid Equation

cfcfcf xxxt

xdxd

ydyxFxd

ygV

xd

yd

V

sdtdJ

002

0

2

00

,,2

1

Minimization Problem

Solution of the Brachistochrone Problem:

SOLO

31

HISTORY OF CALCULUS OF VARIATIONSThe brachistochrone problem

Jacob Bernoulli(1654-1705)

Gottfried Wilhelmvon Leibniz(1646-1716)

Isaac Newton(1643-1727)

The solutions of Leibniz, Johann Bernoulli, Jacob Bernoulliand Newton were published on May 1697 publication ofActa Eruditorum. L’Hôpital solution was published only in 1988.

Guillaume FrançoisAntoine de L’Hôpital

(1661-1704)

SOLO

32

SOLOJohan Bernoulli (continue – 5)



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

SOLODifferential Geometry in the 3D Euclidean Space

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.


Jean Le Rond D’Alembert1717 - 1783

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 bya 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 Bernoulli1700 - 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

http://upload.wikimedia.org/wikipedia/commons/d/de/Danielbernoulli.jpg

36

Daniel Bernoulli



1724 - Mathematical Exercises was published, with Goldbach's assistance,. This consisted of four separate partsThe 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 designedan 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.


SOLO


37

Daniel Bernoulli



1725 - 33 Daniel Bernoulli stayed at St Petersburg.

Leonhard Euler1707 - 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

00

0 & 0x

x

uu

x

(fixed end)

2 3

2 30 & 0

x L x L

u u

x x

(free end)

2

2

uEI F

x x

Fixed end Free end

ux

F

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


http://upload.wikimedia.org/wikipedia/commons/6/60/Leonhard_Euler_2.jpg

38

Daniel Bernoulli



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

vg h p const

Bernoulli’s Principle

http://en.wikipedia.org/wiki/Bernoulli%27s_principle

SOLO


http://upload.wikimedia.org/wikipedia/commons/a/a8/BernoullisLawDerivationDiagram.png

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

http://plus.maths.org/issue1/bern/solutio.gif

http://www.bun.kyoto-u.ac.jp/phisci/Gallery/D.bernoulli_note.html

http://en.wikipedia.org/wiki/1738

http://en.wikipedia.org/wiki/Pierre-Simon_Laplace

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

oK

kg m/ 3

2/ mN

ROBERT BOYLE )1660( DISCOVERED EXPERIMENTALLY THAT THEPRODUCT 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, andIts Effects )1660(

p

V

constT

41

KINETIC THEORY OF GASES

HISTORY

RICHARD TOWNLEY )1628-1707( AND HENRY POWER )1623-1668( FOUNDALSO EXPERIMENTALLY THE P*V LAW IN 1660.

SOLO

STATE EQUATION )BOYLE-MARIOTTE LAW(

EDMÉ MARIOTTE )~1620-1684( INDEPENDENTLY FINDS THERELATIONSHIP 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 PRESSURECHANGES PROPORTIONALLY TO THE SQUARE OFPARTICLE VELOCITIES AS TEMPERATURE CHANGES.

42

Daniel Bernoulli



1733 - He left St Petersburg with his younger brother Johann(II) Bernoulli who was also with him in St Petersburg and they, making visits toDanzig, Hamburg, Holland and Paris before returning to Basel in 1734

Johann II1710-1790

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



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 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



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 Newton1643 - 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.


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 II1710-1790He 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 III1744-1807

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 II1759-1789

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

April 9, 2023 56

SOLO

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 – 2013

Stanford University1983 – 1986 PhD AA

Bernoulli Family

Science

Transcript of Bernoulli Family