The Bernoullis

Champions of Swiss‐German Calculus Jacob and Johann Bernoulli

• Two preeminent mathematicians of the 18th century. • Brothers and members of a family of mathematicians. • Instrumental in the spread and development of calculus. Bernoulli Family Tree

Nicolaus Bernoulli 1623‐1708

Jacob Nicolaus Johann Bernoulli Bernoulli Bernoulli 1654‐1705 1662‐1716 1667‐1748

Nicolaus I Nicolaus II Daniel Johann II Bernoulli Bernoulli Bernoulli Bernoulli 1687‐1759 1695‐1726 1700‐1782 1710‐1790

Johann III Daniel II Nicolaus IV Jakob II Bernoulli Bernoulli Bernoulli Bernoulli 1744‐1807 1751‐1834 1754‐1841 1759‐1789

Nicolaus 1793‐1876

Fritz Theodor 1824‐1913 1837‐1909

Maria Hans Benno Bernoulli Bernoulli Hermann 1868‐1963 1876‐1959 Hesse 1877‐1962 Bernoulli Family

• Originally from Belgium, Jacob and Johann’s grandparents fled from the Netherlands at the time of the Council of Troubles. – King Philip of Spain sent the Duke of Alba to the Netherlands with a sizable army to reaffirm Spanish rule, enforce adherence to Roman Catholicism, and reinforce Philip’s authority. – The Duke set up a court, called the Council of Troubles, that condemned over 12,000 people. Most, like the Bernoullis, fled. Jacob Bernoulli

• Jacob’s father Nicolaus hoped he would take over the family business (Nicolaus was a spice‐ merchant like his father before him). • Instead, Jacob took a degree in theology but eventually made mathematics his life’s work. • Jacob became chair of mathematics at University of Basel. • Had two children, neither of whom became mathematicians. Jacob Bernoulli

• “[Jacob] Bernoulli …was self‐willed, obstinate, aggressive, vindictive, beset by feelings of inferiority, and yet firmly convinced of his own abilities. With these characteristics, he necessarily had to collide with his similarly disposed brother. He nevertheless exerted the most lasting influence on the latter.” ‐‐ J E Hofmann, Biography of Jacob Bernoulli in Dictionary of Scientific Biography (New York 1970‐1990). Jacob Bernoulli

• Ars Conjectandi – The Art of Conjecture. – Cardano, Fermat/Pascal, then Bernoulli. • Posed, but didn’t solve, the catenary problem. • Gave an analytic (calculus‐based) proof of the Tautochrone problem, solved geometrically by Huygens. • Published his own (different) proof of the divergence of the harmonic series along with his brother’s proof. Johann Bernoulli

• Also prepared for a future in the spice trade, Johann ended up studying medicine before settling down to do mathematics. He studied mathematics under his brother Jacob while earning his degree. • Met and taught the Marquis de l’Hôpital, who paid him handsomely for both lessons and for “any discoveries.” The lessons continued through correspondence, and the lessons Johann sent became the basis for l’Hôpital’s calculus textbook. Johann Bernoulli

• Took a post as chair of mathematics at Gröningen in the Netherlands, then assumed Jacob’s post as chair in Basel upon his death. • Three of his children, Nicolaus II, Daniel, and Johann II, went on to become mathematicians. • Daniel’s work Hydrodynamica became a classic text in the field, far outshining Johann’s Hydraulica. • Johann, by the way, tried to pre‐date his book to appear as though it came out before his son’s work, and even suggested his son had plagiarized. Johann Bernoulli

• Leibniz’ Bulldog • Wrote most or all of l’Hôpital’s Analysis of the Infinitely Small (unknowingly). • Proved the divergence of the harmonic series (though the proof appeared in Jacob’s work). • Posed and solved the Brachistachrone problem. • Solved the Catenary Problem posed by Jacob. Episode –The Catenary

• Jacob proposed the Catenary problem: What curve is the representation of a cable hung between two posts, which is acted upon only by it own weight? • Jacob didn’t solve it. • Johann did. • And rubbed it in. Episode –The Catenary

“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… . It is true that it cost me study that robbed me of rest for an entire night … 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 and prove the identity of the catenary with the parabola, since it is entirely false.” Episode –The Catenary

• The solution to the catenary problem is the ௘ೣା௘షೣ hyperbolic cosine function, . ଶ • Johann didn’t have access to this function, but was able to specify it as an integral, and construct it by means of conics. l’Hôpital and Johann Bernoulli

• Although Johann rightly claimed that l’Hôpital’s book was plagiarized from Bernoulli’s “correspondence course” notes, his claims were generally ignored. • l’Hôpital had in fact paid Johann for his work, and thus in some sense owned it. l’Hôpital and Johann Bernoulli

• Proof of Johann’s claims was found in 1922, when a copy of the course notes made by his nephew Nicolaus I was discovered. The notes are virtually identical to l’Hôpital’s calculus text, although l’Hôpital had corrected many errors in the notes. • So, l’Hôpital’s Rule is in fact Bernoulli’s Rule. • l’Hôpital was not, however, a bad mathematician. A Word about Series

• Huygens challenged Leibniz to find the sum of the reciprocals of the triangular numbers: A Word about Series

• Liebniz was clever, fast, and loose. He divided everything by 2: A Word about Series

ଵ ଵ He then noticed that , and ଶ ଶ ଵ ଵ ଵ ଵ ଵ ଵ ଵ ଵ ଵ , and , and , ଺ ଶ ଷ ଵଶ ଷ ସ ଶ଴ ସ ହ and so on. A Word about Series

• Thus, he got ଵ ଵ ଵ ଵ ଵ ଵ ଵ ଵ ଶ ଶ ଶ ଷ ଷ ସ ସ ହ

Removing the parentheses and cancelling with ଵ reckless abandon, he obtained , so . ଶ Which is correct. But by our standards, his method is extremely dangerous. The Harmonic Series

• Both Jacob and Johann proved that the ଵ harmonic series ஶ diverged. Johann ௡ୀଵ ௡ proved it first, which Jacob acknowledged when he published both proofs in a paper. Episode: The Brachistochrone

• Johann proposed the Brachistochrone Problem: Find the curve representing the path that will allow a ball to roll down it in the minimum time. • Solution: The (upside down) cycloid. The Brachistochrone

Johann issued this challenge: • “Let who can seize quickly the prize which we have promised to the solver. Admittedly this prize is neither of gold nor silver, for these appeal only to base and venal souls. … Rather, since virtue itself is its own most desirable reward and fame is a powerful incentive, we offer the prize, fitting for the man of noble blood, compounded of honor, praise, and approbation …” The Brachistochrone

The challenge, part 2: • “… so few have appeared to solve our extraordinary problem, even among those who boast that through special methods … they have not only penetrated the deepest secrets of geometry but also extended its boundaries in marvelous fashion; although their golden theorems, which they imagine known to no one, have been published by others long before.” The Brachistochrone

• Just in case the challenge wasn’t clear enough, he sent a copy of the challenge off to Newton. • In 1697 when this challenge was sent, Newton was involved in running the Royal Mint. But when he received the letter, after coming home “very much tired” at 4 p.m., he “did not sleep till he had solved it, which was by four in the morning.” (The words of his niece.) The Brachistochrone

• Newton remarked, “I do not love … to be … teezed by forreigners about Mathematical things.” • He sent his solution, unsigned, back to Johann. The Brachistochrone

• From William Dunhams’ Journey Through Genius: “There is a legend — probably of dubious authenticity but nonetheless of great charm — that Johann, partially chastened, partially in awe, put down the unsigned document and knowingly remarked, ‘I recognize the lion by his paw.’” The Brachistochrone

• Beyond his own solution and Newton’s, Johann received solutions from Liebniz, L’Hopital, and his brother Jacob. Episode: The Isoperimetric Problem

• In his published paper on the Brachistochrone, Jacob used more general methods than Johann, and in fact posed three other problems which could be solved by his methods. One was the Isoperimetric Problem; to find the closed plane curve with a given perimeter that will have the greatest area. • He challenged Johann by name, and offered an award of 50 ducats if he could solve a particular isoperimetric problem by year’s end. Isoperimetric Problem

• Johann provided a solution and claimed the prize, but had oversimplified the problem and so had an incomplete solution. • Jacob criticized him unmercifully, but could not really revel in it because Johann’s solution had somehow disappeared, not to reappear for publication until after Jacob’s death. • It is generally agreed that Jacob’s solution was superior. Calculus of Variations

• What is now called the calculus of variations grew out of the work the Bernoullis did on the brachistochrone, isochrone, and similar problems. It is the study of finding functions that satisfy optimal conditions. Ars Conjectandi – The Art of Conjecture • Jacob’s masterpiece on probability, published after his death. • The third great advancement in probability theory – Cardano, the Fermat/Pascal correspondences, and then Jacob Bernoulli. Ars Conjectandi

• Nicolaus II, Johann’s oldest son was studying with Jacob and read the nearly‐completed manuscript of Ars Conjectandi. After Jacob died, in true family form, Nicolaus used part of it for his own thesis, among other things. • In 1713, after admitting he was too young and inexperienced to do much with the manuscript, he gave it to the printers with a brief preface, and it was published as Jacob left it. (Weak) Law of Large Numbers

• With virtual certainty, the sample averages will converge to the expected value. That is, if you take samples and compute the average of each sample, the sequence ଵ ଶ ଷ will converge to the expected value. The Brothers Bernoulli

• There is still some historical debate as to which brother accomplished what. Despite their differences, they often worked together, so it is difficult to unravel precedence and authorship of all their ideas. It is also clear that neither brother was above stretching the truth a little to be seen as superior to the other. The Brothers Bernoulli

• It is clear, for example, that the calculus of variations came out of their work, but it is unclear which contributed more to its development. • If there is a consensus, it seems to be that Jacob’s results were deeper and more general, while Johann’s were obtained more quickly and were “clever,” relying on tricks or special attributes of a particular curve or situation.