Lecture 29. Bernouilli Brothers

Figure 29.1 Jacob Bernoulli and Johann Bernouli

The Bernoulli brothers joined Leibniz after 1687. Before 1700 these men hadfound most of our undergraduate calculus, including the calculus of variation.

Bernoulli brothers The ancestors of the Bernoulli family originally lived in Holland. In1583, they migrated to Switzerland and settled at Basel. Bernoulli family is an extraordi-nary Swiss family from Basel that produced eight outstanding mathematicians within threegenerations. Jacob(1654-1705) and Johann (1667-1748) are considered the most importantfounders of calculus (with the exception of Newton and Leibniz).

Following his fathers wish, Jacob studied theology and entered the ministry. But hedeveloped a passion for mathematics in 1676. Jacob Bernoulli basically taught himself inthese subjects and went on to lecture in experimental physics at the University of Basel.Jacob Bernoulli became a professor of mathematics in the same university in 1687.

Around the time, his younger brother Johann, meeting the wishes of his father to go intocommerce, enrolled at the university to study medicine. However, Johann secretly studied

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mathematics with his brother Jacob. Just over two years, Johanns mathematical level wasclose to his brothers.

In 1684, Leibnizs most important work on calculus published. Jacob and Johann quicklyrealized the importance and significance of this work, and began to become familiar withcalculus through a correspondence with Gottfried Leibniz. At the time, Leibnizs publica-tions on the calculus were very obscure to mathematicians and the Bernoulli brothers werethe first to try to understand and apply Leibnizs theories.

Jacobs contributions to calculus Jacob Bernoulli published five treatises on infi-nite series between 1682 and 1704. The first two of these contained many results, such asfundamental result that

1 +1

2+

1

3+ ...+

1

+ ...

diverges, which Bernoulli believed were new but they had actually been proved. Bernoullialso showed that

1 +1

22+

1

32+ ...+

1

2+ ...

converged to a finite limit less than 2. Euler was the first to find the sum of this series, 2

6,

in 1737. Bernoulli also studied the exponential series.

Figure 29.2 Lemniscate of Bernoulli is defined by (2 + 2)2 = 22(2 2).

In 1690 Jacob Bernoulli showed that the problem of determining the isochrone is equiv-alent to solving a first-order non-linear differential equation. The isochrone, or curve ofconstant descent, is the curve along which a particle will descend under gravity from anypoint to the bottom in exactly the same time, no matter what the starting point. It hadbeen studied by Huygens in 1687 and Leibniz in 1689. After finding the differential equa-tion, Bernoulli then solved it by what we now call separation of variables. Jacob Bernoullispaper of 1690 is important for the history of calculus, since the term integral appears forthe first time with its integration meaning.

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In 1696 Bernoulli solved the equation, now called the Bernoulli equation, = ()+().

Johanns contributions to calculus Johann Bernoulli, twelve and a half years youngerthan his brother, lived to be eighty years old. His work was closely related to that of hisolder brother.

He applied calculus on differential equations, finding the lengths of curves and the areasenclosed by curves, isochronous curves, and curves of fastest descent.

Both Jacob and Johann were the inventors of the calculus variations because of theircontribution to the problem of the brachistochrone. 1 The answer to the problem is the cy-cloid. 2 Huygens discovered the cycloid and used it in constructing tautochronous pendulumclocks (1673) in which the period is independent of the amplitude.

Figure 29.3 Brachistochrone curve.

After Jacobs death, Johann Bernoulli took over his chair in University of Basel, where heremained for forty-three more years. After 1700, he mainly concerned himself with problemsof mechanics. After Leibniz died and Newton retreated from scientific work, he was regardedas Europes leading mathematics.

Johann Bernoullis most important student was Leonhard Euler (see next lecture).

Dispute between Jacob and Johann Johann and Jakob two had challenged to eachothers on mathematics and also had bitter arguments about the quality of each others work.Irked by Johanns bragging, Jacob publicly claimed that his younger brother had copied

1Brachistochrone curve, a curve of fastest descent, is the curve between two points that is covered in theleast time by a body that starts at the first point with zero speed and is constrained to move along thecurve to the second point, under the action of constant gravity and assuming no friction.

2A cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rollsalong a straight line. It is an example of a roulette, a curve generated by a curve rolling on another curve.

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his own results. As the chairman of mathematics at Basle, Jacob succeeded in blocking hisbrothers appointment to the same department, forcing him to take a teaching job at theUniversity of Groningen instead.

Figure 29.4 University of Basel

Standing with Leibniz Jacob and Johann brothers were firmly standing with Leibnizto challenge Newton. To find out how much Newton really knew, John Bernoulli, publishedin June 1696 a challenging problem, which he addressed to acutest mathematicians of theworld. Leibniz and Bernoulli were confident that only a person who knows calculus couldsolve this problem.

On 29 January 1697 Newton returned home from working and found the problem inhis post that Bernoulli had sent to him directly. Newton solved the problem. AlthoughNewtons solution was anonymous, he was recognized by Bernoulli as its author; Johannsaid: we know the lion by his claw.

Johann and LHopitals rule Johann instructed lHoospital, who later incorporatedsome of his Johanns results in a textbook he wrote, so that it has come to be known aslHospitals rule. lHospital paid a large monthly salary to exchange for Bernoullis teachingand results. This was the first textbook on calculus, lAnalyse des Infiniment Petits pourlIntelligence des Lignes Courbes. Johann Bernoulli was unhappy that his work was beingpublished by another with only a bare acknowledgement. He kept silent for some times, andeventually published his own lectures on that material.

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Figure 26.8 Daniel Bernoulli

Daniel Bernoulli Johann has three sons and Daniel Bernoulli (1700-1782) is the mostmathematically talented one among the three. Sadly, Johann repeated his fathers mistakeand tried to force Daniel into a career as a merchant, which he didnt want. When theattempt failed, Johann only allowed Daniel to study medicine so that his son would notbecome his competitor. While studying medicine, Daniel took lessons in mathematics fromhis older brother Nikolaus II. When he was 24, Denial was offered a professor position atthe Academy of Science in St. Petersburg.

He returned to Basle in 1734 to discover that he and his father both had been jointlyawarded a prize by the French Academy of Sciences. His father became angry and wasunhappy to admit that his son was at least his equal. As a result, Johann kicked him of thefamily house.

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