The Bernoulli Brothers and the Calculus

1
The Bernoulli Brothers and the Calculus

• V. Frederick Rickey
• USMA, West Point

2
The Bernoulli BrothersJacob I
Johann I 1654-1705 1667-1748 
3
Jacob Bernoulli 1654 - 1705

• The only image before 1700
• Painted by his younger brother Nicholas, 1662
  1716.
• In the Alta Aula, Basel

4
Where is Basel?
5
Jacob Bernoullis Life

• Born, Basel 1654, 5th child
• MA, philosophy, 1671 at University of Basel
  (founded 1460)
• Licentiate in theology, 1676
• Traveled to Geneva, Paris, England, Netherlands
• 1683 back to Basel. Taught experimental
  philosophy
• 1687 Professor of Mathematics at Basel
• Died 1705

6
What did Jacob Bernoulli read?

• René Descartes
• by Lucien Butavand
• after Frans Hals

7
(No Transcript)
8
Descartess Geometry, 1637, 1659
9
John Wallis 1616 - 1703

• Arithmetica infinitorum (1655)
• Tractatus duo (1659)

10
Isaac Barrow 1630-1677

• Lectiones Opticae (1669)
• Lectiones Geometricae (1670)

11
Leibniz Nova methodus of 1684
12
Jacob Bernoullis early work

• Theory of series
• Probability Law of large numbers
• The isochrone problem
• The catenary
• The logarithmic spiral
• Curvature
• Elastica

13
The Law of Large Numbers

• P( Sn/n p lt e ) ? 1 as n ? 8

14
The Isochrone Problem

• Find a curve along which a body will descend
  equal distances in equal times
• He reduces it to the Differential Equation va
  dx vy dy.
• Et eorum integralia !
• The curve is a semi-cubical parabola,
• y3 9/4 a x2

15
(No Transcript)
16
(No Transcript)
17
The Bernoullis on Problem Solving

• Always attack a special problem. If possible
  solve it in a way that leads to a general method.
  
• Read and digest every earlier attempt at a theory
  of the phenomenon in question. Perpend with
  utmost scruple the partial successes and failed
  attempts of the great masters of the past.
• Let a key problem solved be father to a key
  problem posed.
• If two special problems already solved seem
  cognate, try to unite them in a general scheme.
  
• Never rest content with an imperfect or
  incomplete argument. If you cannot complete it
  yourself, lay bare its flaws for others to see.
  
• Never abandon a problem you have solved. There
  are always better ways. Keep searching for them,
  for they lead to fuller understanding. While
  broadening, deepen and simplify.
• Thanks to Clifford Truesdell (1918-2000)

18
Jacob Bernoullis Opera, 1744
19
Johann Bernoulli in 1743

• His spirit sees truth
• His heart knows justice
• He is an honor to the Swiss
• And to all of humanity
• Voltaire

20

• It was a weakness of Voltaire'sTo forget to say
  his prayers,And one which to his shameHe never
  overcame.
• Edmund Clerihew Bentley (1875-1956)

21
Johann Bernoullis Life

• Born, Basel, 1667, 10th child.
• Entered university, age 15, to study business
• Studied mathematics with Jacob
• MA 1685 in experimental physics
• Traveled, 1691, to Geneva, France
• Doctor of medicine, 1694
• Prof at Groningen, 1695-1705
• Back to Basel, 1705
• Died 1748

22
Where is Groningen?
23
Guillaume François Antoine l'Hospital 1661-1704
24
LHospitals Rule
25
(No Transcript)
26
LHospitals Proof of His Rule

• BD bd
• bf / bg
• df /dg
• (df/dx) / (dg/dx)
• f '(a) / g'(a)

27
To the sharpest mathematicians now flourishing
throughout the world

• We are well assured that there is scarcely
  anything more calculated to rouse noble minds to
  attempt work conductive to the increase of
  knowledge than the setting of problems at once
  difficult and useful, by the solving of which
  they may attain to personal fame as it were by a
  specially unique way, and raise for themselves
  enduring monuments with posterity. For this
  reason, I . . . propose to the most eminent
  analysts of this age, some problem, by means of
  which, as though by a touchstone, they might test
  their own methods, apply their powers, and share
  with me anything they discovered, in order that
  each might thereupon receive his due meed of
  credit when I publicly announce the fact.
• 1697

28
To the sharpest mathematicians now flourishing
throughout the world

• To determine the curved line joining two given
  points, situated at different distances from the
  horizontal and not in the same vertical line,
  along which a mobile body, running down by its
  own weight and starting to move from the upper
  point, will descend most quickly to the lowest
  point.

29

• The efforts of my brother were without success
  for my part, I was more fortunate, for I found
  the skill (I say it without boasting, why should
  I conceal the truth?) to solve it in full and to
  reduce it to the rectification of the parabola.
  It is true that it cost me study that robbed me
  of rest for an entire night. It was much for
  those days and for the slight age and practice I
  then had, but the next morning, filled with joy,
  I ran to my brother, who was still struggling
  miserably with this Gordian knot without getting
  anywhere, always thinking like Galileo that the
  catenary was a parabola. Stop! Stop! I say to
  him, don't torture yourself any more to try to
  prove the identity of the catenary with the
  parabola, since it is entirely false. The
  parabola indeed serves in the construction of the
  catenary, but the two curves are so different
  that one is algebraic, the other is
  transcendental.

30
(No Transcript)
31

• With justice we admire Huygens because he first
  discovered that a heavy particle traverses a
  cycloid in the same time, no matter what the
  starting point may be. But you will be struck
  with astonishment when I say that this very same
  cycloid, the tautochrone of Huygens, is the
  brachistochrone we are seeking.
• Johann Bernoulli

32
Christiaan Huygens 1629-1695Horologium
oscillatorium, 1673
33

• Daniel Bernoulli born 1700
• In the Aula at Groningen (founded 1614)

34
Daniel Bernoulli 1700 - 1782

• Won ten prizes from the Paris Academy
• Famous for a work on hydrodynamics

35
(No Transcript)
36
History of LHospitals Rule

• 1696 The rule was published
• 1705 Johann Bernoulli iterates my rule
• 1743 JBs integral calculus published
• 1922 Manuscript of differential calculus found
• 1955 Bernoulli LHospital correspondence
  published

37
(No Transcript)
38
Using Differential Calculus to Resolve Problems,
1691

• A manuscript page of Johann Bernoullis lectures
  on the differential calculus
• The handwriting is that of Nicolaus I Bernoulli,
  1705

39
L'Hospital in Paris to Bernoulli in Basel, 17
March 1694

• I shall give you with pleasure a pension of
  three hundred livres, which will begin on the
  first of January of the present year . . . . I
  promise to increase this pension soon, since I
  know it to be very moderate . . . I am not so
  unreasonable as to ask for this all your time,
  but I shall ask you to give me occasionally some
  hours of your time to work on what I shall ask
  you and also to communicate to me your
  discoveries, with the request not to mention them
  to others. I also ask you to send neither to M.
  Varignon nor to others copies of the notes that
  you let me have, for it would not please me if
  they were made public. Send me your answer to all
  this and believe me, Monsieur tout â vous
  
• le M. de Lhospital

40

• In calculus, de L'Hospital
• Could hardly cope at all.
• Being rich, as a rule he
• Bought results from Bernoulli
• A clerihew by Ralph P. Boas, Jr

41

• Johann Bernoullis best student !
• Leonhard Euler,
• 1707-1783

42
Finding areas under curves

• Decompose the region into infinitely many
  differential areas
• with parallel lines
• with lines emanating from a point
• with tangent lines
• with normal lines.

43
We seek the curve where the square of the
ordinate BC is the mean proportional between the
square of the given length E and the curvilinear
figure ABC.

• E2 / BC2 BC2 / Area ABC
• Area ABC y4 / a2
• By FTC,
• y dx 4 y3 dy / a2
• Divide by y and integrate
• To get a cubical parabola

44

• Obituary of Jacob Bernoulli mentioned the Law of
  Large Numbers
• Montmort tried to prove it and published a book
  on probability in 1708
• Nikolaus I Bernoulli corresponded with Montmort
• So Nikolaus I published the Ars conjectandi in
  1713.

45

• The Art of Conjecturing, just appeared in English
  translation, edited by Edith Sylla.

46
Brook Taylor 1685 - 1731 Methodus
incrementorum, 1715

• Contained work published by Bernoulli
• Only credited Newton
• Johann Bernoulli was incensed !

47

• My God, what does that writer intend by that
  feigned obscurity in which he cloaks matters
  extremly clear by their very nature? No doubt in
  order to conceal his zeal for stealing . . .
  there is nothing in the book except what he has
  stolen from us.
• Johann to Leibniz, 1716

48

• When Taylor died in 1731 at the age of 42 Johann
  Bernoulli gloated
• Taylor is dead. It is a kind of fate that my
  antagonists died before me, all younger than I.
  He is the sixth one of them to die in the last
  fifteen years . . . All these men attacked and
  harassed me . . . though I did them no wrong. It
  seems that heaven would avenge the wrong they
  have done me.

49
Lenore Feigenbaum

• A
• Happy
• Ending !

50

• Château de Montmort, 1990
• Rene B came from Basel
• A Taylor brought olive branches from England
• A Montmort supplied champaigne