A

Fermat’s Principle of Least Time

Variational Principle:

Of all conceivable happenings, nature chooses the one of least ‘effort’.

Principle of Least Action ( Pierre-Louis Moreau de Maupertuis , 1746 )

Hamilton’s Principle

History topic: The

brachistochrone problem

The brachistochrone problem was posed by Johann Bernoulli in Acta

Eruditorum in June 1696. He introduced the problem as follows:-

I, Johann Bernoulli, address the most brilliant mathematicians in the

world. Nothing is more attractive to intelligent people than an

honest, challenging problem, whose possible solution will bestow

fame and remain as a lasting monument. Following the example set

by Pascal, Fermat, etc., I hope to gain the gratitude of the whole

scientific community by placing before the finest mathematicians of

our time a problem which will test their methods and the strength of

their intellect. If someone communicates to me the solution of the

proposed problem, I shall publicly declare him worthy of praise.

The problem he posed was the following:-

Given two points A and B in a vertical plane, what is the curve

traced out by a point acted on only by gravity, which starts at A and

reaches B in the shortest time.

Perhaps we are reading too much into Johann Bernoulli's references

to Pascal and Fermat, but it interesting to note that Pascal's most

famous challenge concerned the cycloid, which Johann Bernoulli

knew at this stage to be the solution to the brachistochrone

problem, and his method of solving the problem used ideas due to

Fermat.

Johann Bernoulli was not the first to consider the brachistochrone problem. Galileo in

1638 had studied the problem in 1638 in his famous work Discourse on two new

sciences. His version of the problem was first to find the straight line from a point A

to the point on a vertical line which it would reach the quickest. He correctly

calculated that such a line from A to the vertical line would be at an angle of 45

reaching the required vertical line at B say.

He calculated the time taken for the point to move from A to B in a straight

line, then he showed that the point would reach B more quickly if it travelled along

the two line segments AC followed by CB where C is a point on an arc of a circle.

Although Galileo was perfectly correct in this, he then made an error

when he next argued that the path of quickest descent from A to B

would be an arc of a circle - an incorrect deduction.

Returning to Johann Bernoulli he stated the problem in Acta

Eruditorum and, although knowing how to solve it himself, he

challenged others to solve it. Leibniz persuaded Johann Bernoulli to

allow a longer time for solutions to be produced than the six months

he had originally intended so that foreign mathematicians would

also have a chance to solve the problem. Five solutions were

obtained, Newton, Jacob Bernoulli, Leibniz and de L'Hôpital solving

the problem in addition to Johann Bernoulli.

Now Johann Bernoulli and Leibniz deliberately tempted Newton with

this problem. It is not surprising, given the dispute over the calculus,

that Johann Bernoulli had included these words in his challenge:-

...there are fewer who are likely to solve our excellent problems,

aye, fewer even among the very mathematicians who boast that

[they]... have wonderfully extended its bounds by means of the

golden theorems which (they thought) were known to no one, but

which in fact had long previously been published by others.

According to Newton's biographer Conduitt, he solved the problem

in an evening after returning home from the Royal Mint. Newton:-

... in the midst of the hurry of the great recoinage, did not come

home till four (in the afternoon) from the Tower very much tired, but

did not sleep till he had solved it, which was by

four in the morning.

Newton sent his solution to Charles Montague, the

Earl of Halifax, who was an innovative finance

minister and the founder of the Bank of England.

Montague was the principal patron and lifelong

friend of Newton and, in addition, the 'common-

law' husband of Newton's niece. He was President of the Royal

Society during the years 1695 to 1698 so it was natural that Newton

send him his solution to the brachistochrone problem. However, as

he wrote afterwards, the episode did not please Newton:-

I do not love to be dunned [pestered] and teased by foreigners

about mathematical things ...

The Royal Society published Newton's solution anonymously in the

Philosophical Transactions of the Royal Society in January 1697. His

solution was explained to Montague as follows:-

Problem. It is required to find the curve ADB in which a weight, by

the force of its gravity, shall descend most swiftly from any given

point A to any given point B.

Solution. From the given point A let there be drawn an unlimited

straight line APCZ parallel to the horizontal, and on it let there be

described an arbitrary cycloid AQP meeting the straight line AB

(assumed drawn and produced if necessary) in the point Q, and

further a second cycloid ADC whose base and height are to the base

and height of the former as AB is to AQ respectively. This last

cycloid will pass through the point B, and it will be that curve along

which a weight, by the force of its gravity, shall descend most

swiftly from the point A to the point B.

The May 1697 publication of Acta Eruditorum contained Leibniz's

solution to the brachistochrone problem on page 205, Johann

Bernoulli's solution on pages 206 to 211, Jacob Bernoulli's solution

on pages 211 to 214, and a Latin translation of Newton's solution on

page 223. The solution by de L'Hôpital was not published until 1988

when, nearly 300 years later, Jeanne Peiffer presented it as

Appendix 1 in [1]. Johann Bernoulli gave the solvers, saying:-

... my elder brother made up the fourth of these, that the three

great nations, Germany, England, France, each one of their own to

unite with myself in such a beautiful search, all finding the same

truth.

Johann Bernoulli's solution divides the plane into strips and he

assumes that the particle follows a straight line in each strip. The

path is then piecewise linear. The problem is to determine the angle

of the straight line segment in each strip and to do this he appeals

to Fermat's principle , namely that light always follows the shortest

possible time of travel. If v is the velocity in one strip at angle a to

the vertical and u in the velocity in the next strip at angle b to the

vertical then, according to the usual sine law

v/sin a = u/sin b.

In the limit, as the strips become infinitely thin, the line

segments tend to a curve where at each point the angle

the line segment made with the vertical becomes the angle the

tangent to the curve makes with the vertical. If v is the velocity at

(x, y) and a is the angle the tangent makes with the vertical then

the curve satisfies

v/sin a = constant.

Now, Galileo had shown that the velocity v satisfies

v = √(2gy)

(where g is the acceleration due to gravity) and substituting for v

gives the equation of the curve as

√y/sin a = constant or y = k sin2a

Use y' = dy/dx = cot a and sin2a = 1/(1+cot2a) = 1/(1+y'2) to get

y(1+y'2) = 2h

for a constant h (= 1/(2k2)).

The cycloid x(t) = h(t - sin t), y(t) = h(1 - cos t) satisfies this

equation. To see this note that

y' = dy/dx = dy/dt . dt/dx = -(sin t)/(1 - cos t)

so

y(1+y'2) = h(1 - cos t)(1+sin2t/(1-cos t)2)

= h(1 - cos t + sin2t/(1-cos t))

= h((1-cos t)2+ sin2t)/(1-cos t)

= h(2-2cos t)/(1-cos t) = 2h

Now Huygens had shown in 1659, prompted by Pascal's challenge

about the cycloid, that the cycloid is the solution to the tautochrone

problem, namely that of finding the curve for which the time taken

by a particle sliding down the curve under uniform gravity to its

lowest point is independent of its starting point.

Johann Bernoulli ended his solution of the brachistochrone problem

with these words:-

Before I end I must voice once more the admiration I feel for the

unexpected identity of Huygens' tautochrone and my

brachistochrone. I consider it especially remarkable that this

coincidence can take place only under the hypothesis of Galileo, so

that we even obtain from this a proof of its correctness. Nature

always tends to act in the simplest way, and so it here lets one

curve serve two different functions, while under any other

hypothesis we should need two curves ...

Despite the friendly words with which Johann Bernoulli described his

brother Jacob Bernoulli's solution to the brachistochrone problem

(see above), a serious argument erupted between the brothers after

the May 1697 publication of Acta Eruditorum. It was Jacob Bernoulli

who now challenged his brother. Returning to Galileo's original

question regarding the time to reach a vertical line rather than a

point he asked:-

Given a starting point and a vertical line, of all the cycloids from the

starting point with the same horizontal base, which will allow the

point subjected only to uniform gravity, to reach the vertical line

most quickly.

Johann Bernoulli solved this problem showing that the cycloid which

allows the particle to reach the given vertical line most quickly is the

one which cuts that vertical line at right angles. There is a wealth of

information in the correspondence with Varignon given in [1]. Jacob

Bernoulli posed isoperimetric problems to Johann Bernoulli and a

bitter dispute arose between the two brothers on these problems

which Varignon also became involved in. It was an unpleasant

incident, but one of great value to mathematics for the problems

being argued about led directly to the founding of the calculus of

variations. The quarrel between the Bernoulli brothers is examined

in detail in [10] where as well as the mathematical details the

author studies the psychological side. He argues convincingly that

the bad feeling between them must a started at home with a strict

and unfriendly father.

The methods which the brothers developed to solve the challenge

problems they were tossing at each other were put in a general

setting by Euler in Methodus inveniendi lineas curvas maximi

minimive proprietate gaudentes sive solutio problematis

isoperimetrici latissimo sensu accepti published in 1744. In this

work, the English version of the title being Method for finding plane

curves that show some property of maxima and minima, Euler

generalises the problems studies by the Bernoulli brothers but

retains the geometrical approach developed by Johann Bernoulli to

solve them. He found what has now come to be known as the Euler-

Lagrange differential equation for a function of the maximising or

minimising function and its derivative.

The idea is to find a function which maximises or minimises a

certain quantity where the function is constrained to satisfy certain

constraints. For example Johann Bernoulli had posed certain

geodesic problems to Euler which, like the brachistochrone problem,

were of this type. Here the problem was to find curves of minimum

length where the curves were constrained to lie on a given surface.

Euler, however, commented that his geometrical approach to these

problems was not ideal and it only gave necessary conditions that a

solution has to satisfy. The question of the existence of a solution

was not solved by Euler's contribution.

Lagrange, in 1760, published Essay on a new method of determining

the maxima and minima of indefinite integral formulas. It gave an

analytic method to attach calculus of variations type problems. In

the introduction to the paper Lagrange gives the historical

development of the ideas which we have described above but it

seems appropriate to end this article by giving what is in effect a

summary of the developments in Lagrange's words:-

The first problem of this type [calculus of variations] which

mathematicians solved was that of the brachistochrone, or the

curve of fastest descent, which Johann Bernoulli proposed towards

the end of the last century. The solution was found by considering

special cases, and it was only some time later, in research

isoperimetric curves, that the great mathematician of whom we

speak and his famous brother Jacob Bernoulli gave some general

rules for solving several other problems of the same type. Since,

however, the rules were not sufficiently general, the famous Euler

undertook the task of reducing all such investigations to a general

method which he gave in the work "Essay on a new method of

determining the maxima and minima of indefinite integral

formulas"; an original work in which the profound science of the

calculus shines through. Even so, while the method is ingenious and

rich, one must admit that it is not as simple as one might hope in a

work of pure analysis ....

Lagrange then goes on to describe his introduction of the differential

symbol . He gives:-

... a method which only requires a straightforward use of the

principles if the differential and integral calculus; but I must strongly

emphasise that since my method requires that a quantity be

allowed to vary in two different ways, so as not to confuse these

different variations, I have introduced a new symbol into my

calculations. In this way z expresses a difference of z which is different from dz, but which, however, will satisfy the same rules;

such that where we have for any equation dz = m dx, we can

equally have z = m x, and likewise in other cases.

Article by: J J O'Connor and E F Robertson

February 2002

MacTutor History of Mathematics

A Brachistochrone curve, or curve of fastest descent, is the curve

between two points that is covered in the least time by a body that

starts at the first point with zero speed and passes down along the

curve to the second point, under the action of constant gravity and

ignoring friction.

The brachistochrone is the cycloid

Given two points A and B, with A not lower than B, there is just one

upside down cycloid that passes through A with infinite slope,

passes also through B and does not have maximum points between

A and B. This is the brachistochrone curve. The brachistochrone thus

does not depend on the body's mass or on the strength of the

gravitational constant.

The problem can be solved with the tools from the calculus of

variations.

Note that if the body is given an initial velocity at A, or if friction is

taken into account, the curve that minimizes time will differ from

the one described above.

[edit] Proof

According to Fermat’s principle: The actual path between two points

taken by a beam of light is the one which is traversed in the least

time. Hence, the brachistochrone curve is simply the trajectory of a

beam of light in a medium where the speed of light increases

following a constant vertical acceleration (that of gravity g). The

conservation law can be used to express the velocity of a body in a

constant gravitational field as:

,

where h represents the altitude difference between the current

position and the starting point. It should be noted that the velocity

does not depend on the horizontal displacement.

According to Snell's law, a beam of light throughout its trajectory

must obey the equation:

,

for some constant K, where θ represents the angle of the trajectory

with respect to the vertical. Inserting the velocity expressed above,

we can draw immediately two conclusions:

1- At the onset, when the particle velocity is nil, the angle must be

nil. Hence, the brachistochrone curve is tangent to the vertical at

the origin.

2- The velocity reaches a maximum value when the trajectory

becomes horizontal.

For simplification purposes, we assume that the particle (or the

beam) departs from the point of coordinates (0,0) and that the

maximum velocity is reached at altitude –D. Snell’s law then takes

the expression:

.

At any given point on the trajectory we have:

.

Inserting this expression in the previous formula, and rearranging

the terms, we have:

.

Which is the differential equation of the opposite of a cycloid

generated by a circle of diameter D.

[edit] History

Galileo incorrectly stated in 1638 in his Two New Sciences that this

curve was an arc of a circle. Johann Bernoulli solved the problem (by

reference to the previously analysed tautochrone curve) before

posing it to readers of Acta Eruditorum in June 1696. Four

mathematicians responded with solutions: Isaac Newton, Jakob

Bernoulli (Johann's brother), Gottfried Leibniz and Guillaume de

l'Hôpital. Three of the solutions (excluding l'Hôpital's) were

published in the May 1697 edition of the same publication.

In an attempt to outdo his brother, Jakob Bernoulli created a harder

version of the brachistochrone problem. In solving it, he developed

new methods that were refined by Leonhard Euler into what the

latter called (in 1766) the calculus of variations. Joseph-Louis de

Lagrange did further work that resulted in modern infinitesimal

calculus.

Another rivalry, between Newton and Leibniz, also contributed to

this development. Each claimed to have solved the brachistochrone

problem before the other, and they continued to quarrel over their

subsequent work on the calculus.

[edit] Etymology

In Greek, brachistos means "shortest" and chronos means "time".

[edit] See also

Calculus of variations

Tautochrone curve

A

REVISITING A

CLASSIC

LEAST TIME

PROBLEMBen Szapiro

The Department of Physics

The University of the South, Sewanee

Abstract

The brachistochrone problem, that is, "find the path of shortest time of a particle moving between two points on a vertical plane", was proposed, solved erroneously, and studied experimentally by Galileo, and solved mathematically by Jacques Bernoulli's variational calculus methods in 1697. We will revisit the brachistochrone problem from the prospective of an undergraduate student: we will analyze the "default" answer (straight line = least time), the possibility of breaking down the motion in a succession of straight lines, the subsequent time optimization strategy, friction and rolling ball effects, the 3-D extension, and its connection with Huygens' cycloidal pendulum. Finally, a demonstration of the apparatus will be presented and compared "live" with a simulation of the motion.

Extremum problems provide wonderful material for teaching thinking, inventiveness, flexibility, creativity... But the only way to teach thinking is with concrete special problems... But we still have to show the existence of general principles and laws in science...But we also want to transmit science as part

of our cultural heritage...But the notation and methods look so different....But...(

"BRACHISTOCHRONE"

BRACHIS = SHORT

CHRONOS = TIME

From Acta Eroditorum, the first

scientific journal (Vol. 15):

Statement of the Problem:

"Let two points A and B be given in

a vertical plane. Find the curve that

a point M, moving on a path AMB

must follow such that, starting from

A, reaches B in the shortest time

under its own gravity"

RESPONSES TO THE CHALLENGE:

1) Johann Bernoulli (Of course;

solved Fermat's way)

2) Gottfried Wilhelm von Leibniz

("Splendid problem")

3) Jakob Bernoulli (Johann's

brother, legal counsel says OK)

4) de l'Hospital

5) Isaac Newton (as "an anonymous

englishman", but: "ex ungue leonem")

(The first Dream Team)

"If one considers motions with the same initial and

terminal points

then the shortest distance between them being a

straight line ,

one might think that the motion along it needs least

time.

It turns out that this is not so."

Galileo Galilei

Discourses on Mechanics (1588)

Johann Bernouilli

(1667-1748)

 

Lets look at the solution

Galileo Galilei (1629-1695) and

Christiaan Huygens (1564-1642)

Straight from A -> B (distance s)

First Down to C, then Horizontal to B:

Bernoulli Family

An extraordinary Swiss family from Basle that produced eight outstanding

mathematicians within three generations. Together with Isaac Newton,

Gottfried Leibniz, Leonhard Euler, and Joseph Lagrange, the Bernoulli family

dominated mathematics and physics in the seventeenth and eighteenth

centuries, making important contributions to differential calculus, geometry,

mechanics, ballistics, thermodynamics, hydrodynamics, optics, elasticity,

magnetism, astronomy, and probability theory. Unfortunately, the Bernoullis

were as conceited and arrogant as they were brilliant, and engaged in bitter

rivalries and rows with one another.

The patriarchs of this mathematical dynasty were Jakob I (1654-1705) and his

brother Johann I (1667-1748). (The Roman numerals are to tell fathers,

brothers, sons and cousins apart, as the same Christian names kept being

used in the family). Next came Jakob's son, Nikolaus I, and Johann's three

sons, Nikolaus II, Daniel (1700-1872), and Johann II. Finally, came Johann II's

mathematical offspring, Johann III and Jakob II.

Jakob I developed a passion for science and mathematics after meeting

Robert Boyle during a trip to England in 1676. He largely taught himself in

these subjects and went on to lecture in experimental physics at the

University of Basle. He also secretly introduced his younger brother to

mathematics, much against the wishes of his parents who wanted the

younger brother to go into commerce. The cooperation between the two

brothers soon degenerated, however, into vitriolic argument. Irked by

Johann's bragging, Jakob publicly claimed that his younger brother had

copied his own results. Later, having been appointed to the chair of

mathematics at Basle, Jakob succeeded in blocking his brother's appointment

to the same department, forcing Johann to take a teaching job at the

University of Groningen instead. Johann proposed the so-called

brachistrochrone problem and, along with Newton, Leibniz, l'Hospital, and

Jakob, managed to solve it – but only after he first came up with a faulty proof

and then tried to substitute one of Jakob's in its place! Eventually, Johann was

offered a post at Basel as, of all things, the department head of Ancient

Greek. But, en route to Basel, Johann learned that Jakob had died of

tuberculosis. Upon his arrival he set about lobbying for the vacant position

and, in less than two months, got his way. Jakob's most important work, his

Ars Conjectandi (The Art of Conjecture), was published posthumously and

formed the basis of probability theory.

Sadly, Johann I repeated his father's mistake and tried to force the most

mathematically talented of his three sons, Daniel, into a career as a merchant,

which he didn't want. When the attempt failed, Johann allowed Daniel to study

medicine, in order to prevent his son from becoming a competitor. But all

three sons followed their father's path and Daniel, while studying medicine,

took lessons in mathematics from his older brother Nikolaus II. In 1720 he

traveled to Venice in order to work as a physician but gained such a great

reputation during his stay for his work in physics and mathematics that that

Peter the Great of Russia offered him a chair at the Academy of Science in St.

Petersburg. Daniel went, along with Nikolaus II, who was also offered a

position at the Academy. However, after just eight months, Nikolaus fell ill with

a fever and died. Distressed, Daniel wanted to return to Basle but Johann I

didn't want his son – a potential rival – back home. Instead he sent one of his

pupils, none other than the great Leonhard Euler, to St. Petersburg to keep

Daniel company. A close friendship developed between the two Swiss

mathematicians in exile and the six years they spent together in St.

Petersburg were the most productive of Daniel's life.

When Daniel finally returned to Basle, quarrels within the family flared up

again after he won the prize of the Parisian Academy of Science with a paper,

produced jointly with his father, on astronomy. Upset by Daniel's success,

Johann kicked him of the family house. And worse was to come. In 1738

Daniel published his magnum opus, Hydrodynamica. Johann I read the book,

hurriedly wrote one of his own with the title Hydraulica, back-dated it to 1732,

and claimed to be the inventor of fluid dynamics! The plagiarism was soon

uncovered, and Johann was ridiculed by his colleagues but his son never

recovered from the blow.

Family Squabbles: The Bernoulli Family

Jacob Bernouli (1654 - 1705) Johann Bernouli (1667 - 1748)Daniel Bernoulli (1700 - 1787)

The Bernoulli family may sound like a Mafia family from a television show,

but they were the most predominant math family of Europe. Their fame

was in the late 17th and early 18th century in Bale, Switzerland. The

uniqueness of this particular family is a stubborn streak which brought

devastation to the family life.

The Bernoulli family was originally from Holland with strong Calvinism

religion. They needed to avoid Spanish religious persecution, so they fled

to Switzerland. Nicholas Bernoulli brought the family to Switzerland. This

family was not math oriented, they had a spice business in Bale. He had

three sons which two of them became the most influential math experts

in the academic community yet hostile to each other.

The eldest son, Jacob (James or Jacques), was born 1654 and died 1705 in

Bale, Switzerland. His parents compelled him to study philosophy and

theology. Like a Bernoulli, he resented the studies but he did acquire a

masters in philosophy. He was intrigued with mathematics and astronomy

so much he included them with his studies, regardless his parents wishes.

He made more of a career in mathematics than philosophy. He became

the first Bernoulli to be recognized as a strong influential mathematician.

He contributed highly to probability: if something is going to happen

again and again is large amounts, it is likely it will happen most of the

time. He may be a brilliant mathematician, but he did have a mean

streak.

His younger brother, Johann or John, was born in 1667 and died 1748 in

Bale Switzerland. He did not do very well in the spice business. At 16, he

entered the University of Bale and studied medicine. He asked his

brother, Jacob, to teach him mathematics. By this time, Jacob was a

professor of mathematics. After two years under Jacob's tutelage, Johann

was his equal.

At first, Jacob had no problem teaching his little brother. He realized his

brother's talents and quick-study of mathematics that he offered to work

with Johann. As time went on, the Bernoulli blood began to boil.

Johann's ego was getting larger which he began to brag about his work

and at the same time he belittled his brother. Jacob was so angry he

made crude comments about Johann's abilities. Jacob refer to him as a

student repeating what the teacher taught him, in other words a parrot.

Jacob and Johann went back and forth with comments in the academic

community which developed a notorious reputation of their family

togetherness.

Despite family problems, Johann was a excellent mathematician. He used

calculus to solve problems which Newton failed to solve in the laws of

gravitaiton. By using y=x2, he made all sorts of discoveries of calculus.

In 1695, Jacob was the chair of mathematics in Bale. Johann wanted that

chair, but he was offered a chair in Holland. He vowed not to come back

to Bale. About 1705, Johann's father-in-law was dying and asking for his

daughter and grandchildren, so Johann came back to Bale. While

traveling, he did not know his brother, Jacob, died of tuberculosis. Once

he realized of his brother's death, Johann took his chair.

Johann had three sons and one of his sons became a profound

mathematician. Daniel Bernoulli, born 1700 and died 1787 in Bale. Johann

was determined to make Daniel a merchant. Like a Bernoulli, Daniel did

not want to learn the business. He wanted to study mathematics. Of

course, Johann's stubborness made Daniel to study medicine. He tried to

convince Daniel that there is no money in mathematics. Daniel did study

medicine and applied mathematical physics to it, which he received a

medical doctorate.

Daniel Bernoulli was natural philosopher who applied mathematics in his

work. He developed Hydrodynamics. He analyzed the flow of water from a

hole in a container. This was for conservation of energy which he

developed pumps and machines to raise water.

Daniel was a home-body person. He did not like to travel much. He would

get sick, complain about the weather, and be miserable. Daniel travel to

Danzig, Hamburg, Holland, and Paris. He worked in Venice and St.

Petersburg mostly. He would ask his father to come home, but his father

said no. His father sent his best student, Leonard Euler, to work with him.

Daniel and Euler worked in St. Petersburg on the vibration and frequency

of sounds by using musical instruments. In 1734, he returned to Bale and

entered a contest in Paris Academy for his ideas of astronomy. His father

entered at the same time which they jointly won the Grand Prize. Johann's

ego could not stand being pronounced as an equal to his son, so he

banned Daniel from his house. Johann went so far as stole one of Daniel's

papers and submit his name to it.

The Bernoulli blood may be filled with fire, but they did have a passion for

mathematics.

THE BERNOULLI FAMILY

    One of the most distinguished

families in the history of

mathematics and science is the

Bernoulli family of Switzerland,

which from the late secenteenth

century on, producced an unusual

number of capable

mathematicians and

scientists.  The family record

starts with the two brothers, Jakob

Bernoulli(1654-1705) and Johann

Bernoulli (1667-1748), some of

whose mathematical

accomplishments have already

been mentioned in this

book.  These two men gave up earlier vocational

interests and became mathematicians when Leibniz'

papers began to appear in the Acta eruditorum.

They were among the first mathematicians to realize

the surprising power of the calculcus and to apply the

tool to a great diversity of problems.  From 1687 until

his death, Jakob occupied the mathematics chair at

Basel University.  Johann, in 1697, became a professor

at Groningen University, and then, on Jakob's death in

1705, succeeded his brother in the chair at Basel

University, to remain there for the rest of his life.   The

two brothers, often bitter rivals, maintained an almost

constant exchange of ideas with Leibniz and with each

other.

     One of the first mathematicians to work in the

calculus of variations.   He was also one of the early

students of mathematical probability; his book in this

field, the Ars conjectandi, was posthumously

published in 1713. Several things in mathematics now

bear Jakob Bernoulli's name. Among these are the

Bernoulli distribution and Bernolli theorem of statistics

and probability theory; the Bernoulli equation, met by

every student of a first course in differential

equations;the Bernoulli numbers, and Bernoulli

polynomials first course in the calculus. In jakob

Bernoulli's soulution to the problem of the isochrone

curve, which was published in the Acta eruditorum, in

1690. we meet for the first time the word integral in a

caculus sense. Leibniz had called the integral calculus

calculus summatorius; in 1696, Leibniz and Johann

Bernoulli agreed to call it calculus calculus integralis.

Jakob Bernoulli was struck by the way the equiangular

spiral reproduces itselt under a bariety of

thransformations and asked, in imitation of

Archimedes, that such a spiral be engraved on his

tombstone, along with the inscription "Eadem mutata

resurgo" ("Though changed, I arise again the same.")

    Johann Bernoulli was an even

more prolific contributor to

mathematics than was his brother

Jakob, Though he was a jealous

and cantankerous man, he was

one of the most successful

teachers of his time. He greatly

enriched the cajculus and was

very influential in making the

power of the new subject

apprecianted in contincntal

Europe. As we have seen, if was

his material that the Marquis de

I;Hospital (1661-1704), under a

curious

financial agreement with Johann, assembled in 1696

into the first calculus textbook. In this way, the

familiar metgod of evaluationg the indeterminate form

0/0 became incorrectly known in later calculus texts

as I'Hospital's rule.     Johann Bernoulli had three sons,

Nicolaus (1695-1726), Daniel(1700- 1782), and Johann

II (1710-1790), all of whom won renown as

eighteenthcentury mathematicians and scientists.

Nicolaus, who showed great promise in the field of

mathematics, was called to the St.Petersburg

Academy, where he unfortunately died by drowining,

only eight months later. He wrote on curves,

differential equations, and probability. 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. Daniel returned to Basel seven years

later. He was the most famous of Johann's three sons,

and devoted most of his energies to probability,

astronmy, physics and hydrodynamics. In probability

he debised the concept of moral expectation, and in

his Hydrodynamica, of 1738, appears the principle of

hydrodynamics that bears his name in all present-day

elementary physics texts.

    Johann II, the youngest of the three sons, 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.

    There was another eighteenth-century Nicolaus

Bernoulli (1687-1759), a nephew of Jakob and Johann,

who achieved some fame in mathematics. This

Nicoiaus held, for a time, the chair of mathematics at

Padua once filled by Galileo. He wrote extensively on

geometry and differential equations. Later in life, he

taught logic and law.

    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.

    LesserBernoulli 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.

A