The Bernoulli Brothers

and the Calculus

V. Frederick Rickey

USMA, West Point

The Bernoulli BrothersJacob I Johann I

1654-1705 1667-1748 

Jacob Bernoulli 1654 - 1705

• The only image before 1700

• Painted by his younger brother Nicholas, 1662 – 1716.

• In the Alta Aula, Basel

Where is Basel?

Jacob Bernoulli’s 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

What did Jacob Bernoulli read?

• René Descartes

• by Lucien Butavand

• after Frans Hals

Descartes’s Geometry, 1637, 1659

John Wallis 1616 - 1703

• Arithmetica infinitorum (1655)

• Tractatus duo (1659)

Isaac Barrow 1630-1677

• Lectiones Opticae (1669)

• Lectiones Geometricae (1670)

Leibniz & Nova methodus of 1684

Jacob Bernoulli’s early work

• Theory of series

• Probability: Law of large numbers

• The isochrone problem

• The catenary

• The logarithmic spiral

• Curvature

• Elastica

The Law of Large Numbers

P( | Sn/n – p | < ε ) → 1 as n → ∞

The Isochrone Problem

• Find a curve along which a body will descend equal distances in equal times

• He reduces it to the Differential Equation √a dx = √y dy.

• Et eorum integralia !

• The curve is a semi-cubical parabola,

y3 = 9/4 a x2

The Bernoullis on Problem Solving1. Always attack a special problem. If possible solve it in a way that leads to a general

method.

2. 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.

3. Let a key problem solved be father to a key problem posed.

4. If two special problems already solved seem cognate, try to unite them in a general scheme.

5. Never rest content with an imperfect or incomplete argument. If you cannot complete it yourself, lay bare its flaws for others to see.

6. 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)

Jacob Bernoulli’s Opera, 1744

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

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)

Johann Bernoulli’s 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

Where is Groningen?

Guillaume François Antoine l'Hospital 1661-1704

L’Hospital’s Rule

L’Hospital’s Proof of His Rule

BD ≈ bd

= bf / bg

≈ df /dg

= (df/dx) / (dg/dx)

= f ′(a) / g′(a)

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

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.

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.

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

Christiaan Huygens 1629-1695Horologium oscillatorium, 1673

• Daniel Bernoulli born 1700

• In the Aula at Groningen (founded 1614)

Daniel Bernoulli 1700 - 1782

• Won ten prizes from the Paris Academy

• Famous for a work on hydrodynamics

History of L’Hospital’s Rule

• 1696 The rule was published

• 1705 Johann Bernoulli iterates “my rule”

• 1743 JB’s integral calculus published

• 1922 Manuscript of differential calculus found

• 1955 Bernoulli – L’Hospital correspondence published

• A manuscript page of Johann Bernoulli’s lectures on the differential calculus

• The handwriting is that of Nicolaus I Bernoulli, 1705

Using Differential Calculus to Resolve Problems, 1691

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

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

• Johann Bernoulli’s best student !

• Leonhard Euler,

1707-1783

Finding areas under curves

Decompose the region into infinitely many differential areas

1. with parallel lines

2. with lines emanating from a point

3. with tangent lines

4. with normal lines.

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

• 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.

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

Brook Taylor 1685 - 1731 Methodus incrementorum, 1715

• Contained work published by Bernoulli

• Only credited Newton

Johann Bernoulli was incensed !

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

• 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.

Lenore Feigenbaum

A

Happy

Ending !

• Château de Montmort, 1990

• Rene B came from Basel

• A Taylor brought olive branches from England

• A Montmort supplied champaigne