October 6, 2020October 8, 2020 by David D. Nolte

The Bountiful Bernoullis of Basel

The task of figuring out who’s who in the Bernoulli family is a hard nut to crack. The Bernoulli name populates a dozen different theorems or physical principles in the history of science and mathematics, but each one was contributed by any of four or five different Bernoullis of different generations—brothers, uncles, nephews and cousins. What makes the task even more difficult is that any given Bernoulli might be called by several different aliases, while many of them shared the same name across generations. To make things worse, they often worked and published on each other’s problems.

To attribute a theorem to a Bernoulli is not too different from attributing something to the famous mathematical consortium called Nicholas Bourbaki. It’s more like a team rather than an individual. But in the case of Bourbaki, the goal was selfless anonymity, while in the case of the Bernoullis it was sometimes the opposite—bald-faced competition and one-up-manship coupled with jealousy and resentment. Fortunately, the competition tended to breed more output than less, and the world benefited from the family feud.

The Bernoulli Family Tree

The Bernoullis are intimately linked with the beautiful city of Basel, Switzerland, situated on the Rhine River where it leaves Switzerland and forms the border between France and Germany . The family moved there from the Netherlands in the 1600’s to escape the Spanish occupation.

The first Bernoulli born in Basel was Nikolaus Bernoulli (1623 – 1708), and he had four sons: Jakob I, Nikolaus, Johann I and Hieronymous I. The “I”s in this list refer to the fact, or the problem, that many of the immediate descendants took their father’s or uncle’s name. The long-lived family heritage in the roles of mathematician and scientist began with these four brothers. Jakob Bernoulli (1654 – 1705) was the eldest, followed by Nikolaus Bernoulli (1662 – 1717), Johann Bernoulli (1667 – 1748) and then Hieronymous (1669 – 1760). In this first generation of Bernoullis, the great mathematicians were Jakob and Johann. More mathematical equations today are named after Jakob, but Johann stands out because of the longevity of his contributions, the volume and impact of his correspondence, the fame of his students, and the number of offspring who also took up mathematics. Johann was also the worst when it came to jealousy and spitefulness—against his brother Jakob, whom he envied, and specifically against his son Daniel, whom he feared would eclipse him.

Jakob Bernoulli (aka James or Jacques or Jacob)

Jakob Bernoulli (1654 – 1705) was the eldest of the first generation of brothers and also the first to establish himself as a university professor. He held the chair of mathematics at the university in Basel. While his interests ranged broadly, he is known for his correspondences with Leibniz as he and his brother Johann were among the first mathematicians to apply Lebiniz’ calculus to solving specific problems. The Bernoulli differential equation is named after him. It was one of the first general differential equations to be solved after the invention of the calculus. The Bernoulli inequality is one of the earliest attempts to find the Taylor expansion of exponentiation, which is also related to Bernoulli numbers, Bernoulli polynomials and the Bernoulli triangle. A special type of curve that looks like an ellipse with a twist in the middle is the lemniscate of Bernoulli.

Perhaps Jakob’s most famous work was his Ars Conjectandi (1713) on probability theory. Many mathematical theorems of probability named after a Bernoulli refer to this work, such as Bernoulli distribution, Bernoulli’s golden theorem (the law of large numbers), Bernoulli process and Bernoulli trial.

Fig. Bernoulli numbers in Jakob’s *Ars Conjectandi* (1713)

Johann Bernoulli (aka Jean or John)

Jakob was 13 years older than his brother Johann Bernoulli (1667 – 1748), and Jakob tutored Johann in mathematics who showed great promise. Unfortunately, Johann had that awkward combination of high self esteem with low self confidence, and he increasingly sought to show that he was better than his older brother. As both brothers began corresponding with Leibniz on the new calculus, they also began to compete with one another. Driven by his insecurity, Johann also began to steal ideas from his older brother and claim them for himself.

A classic example of this is the famous brachistrochrone problem that was posed by Johann in the Acta Eruditorum in 1696. Johann at this time was a professor of mathematics at Gronigen in the Netherlands. He challenged the mathematical world to find the path of least time for a mass to travel under gravity between two points. He had already found one solution himself and thought that no-one else would succeed. Yet when he heard his brother Jakob was responding to the challenge, he spied out his result and then claimed it as his own. Within the year and a half there were 4 additional solutions—all correct—using different approaches. One of the most famous responses was by Newton (who as usual did not give up his method) but who is reported to have solved the problem in a day. Others who contributed solutions were Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus, and Guillaume de l’Hôpital in addition to Jakob.

The participation of de l’Hôpital in the challenge was a particular thorn in Johann’s side, because de l’Hôpital had years earlier paid Johann to tutor him in Leibniz’ new calculus at a time when l’Hôpital knew nothing of the topic. What is today known as l’Hôpital’s theorem on ratios of limits in fact was taught to l’Hôpital by Johann. Johann never forgave l’Hôpital for publicizing the result—but l’Hôpital had the discipline to write a textbook while Johann did not. To be fair, l’Hôpital did give Johann credit in the opening of his book, but that was not enough for Johann who continued to carry his resentment.

When Jakob died of tuberculosis in 1705, Johann campaigned to replace him in his position as professor of mathematics and succeeded. In that chair, Johann had many famous students (Euler foremost among them, but also Maupertuis and Clairaut). Part of Johann’s enduring fame stems from his many associations and extensive correspondences with many of the top mathematicians of the day. For instance, he had a regular correspondence with the mathematician Varignon, and it was in one of these letters that Johann proposed the principle of virtual velocities which became a key axiom for Joseph Lagrange’s later epic work on the foundations of mechanics (see Chapter 4 in Galileo Unbound).

Johann remained in his chair of mathematics at Basel for almost 40 years. This longevity, and the fame of his name, guaranteed that he taught some of the most talented mathematicians of the age, including his most famous student Leonhard Euler, who is held by some as one of the four greatest mathematicians of all time (the others were Archimedes, Newton and Gauss) [1].

Nikolaus I Bernoulli

Nikolaus I Bernoulli (1687 – 1759, son of Nikolaus) was the cousin of Daniel and nephew to both Jacob and Johann. He was a well-known mathematician in his time (he briefly held Galileo’s chair in Padua), though few specific discoveries are attributed to him directly. He is perhaps most famous today for posing the “St. Petersburg Paradox” of economic game theory. Ironically, he posed this paradox while his cousin Nikolaus II Bernoulli (brother of Daniel Bernoulli) was actually in St. Petersburg with Daniel.

The St. Petersburg paradox is a simple game of chance played with a fair coin where a player must buy in at a certain price in order to place $2 in a pot that doubles each time the coin lands heads, and pays out the pot at the first tail. The average pay-out of this game has infinite expectation, so it seems that anyone should want to buy in at any cost. But most people would be unlikely to buy in even for a modest $25. Why? And is this perception correct? The answer was only partially provided by Nikolaus. The definitive answer was given by his cousin Daniel Bernoulli.

Daniel Bernoulli

Daniel Bernoulli (1700 – 1782, son of Johann I) is my favorite Bernoulli. While most of the other Bernoullis were more mathematicians than scientists, Daniel Bernoulli was more physicist than mathematician. When we speak of “Bernoulli’s principle” today, the fundamental force that allows birds and airplanes to fly, we are referring to his work on hydrodynamics. He was one of the earliest originators of economic dynamics through his invention of the utility function and diminishing returns, and he was the first to clearly state the principle of superposition, which lies at the heart today of the physics of waves and quantum technology.

While in St. Petersburg, Daniel conceived of the solution to the St. Petersburg paradox (he is the one who actually named it). To explain why few people would pay high stakes to play the game, he devised a “utility function” that had “diminishing marginal utility” in which the willingness to play depended on ones wealth. Obviously a wealthy person would be willing to pay more than a poor person. Daniel stated

The determination of the value of an item must not be based on the price, but rather on the utility it yields…. There is no doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man though both gain the same amount.

He created a log utility function that allowed one to calculate the highest stakes a person should be willing to take based on their wealth. Indeed, a millionaire may only wish to pay $20 per game to play, in part because the average payout over a few thousand games is only about $5 per game. It is only in the limit of an infinite number of games (and an infinite bank account by the casino) that the average payout diverges.

Johann II Bernoulli

Daniel’s brother Johann II (1710 – 1790) published in 1736 one of the most important texts on the theory of light during the time between Newton and Euler. Although the work looks woefully anachronistic today, it provided one of the first serious attempts at understanding the forces acting on light rays and describing them mathematically [5]. Euler based his new theory of light, published in 1746, on much of the work laid down by Johann II. Euler came very close to proposing a wave-like theory of light, complete with a connection between frequency of wave pulses and colors, that would have preempted Thomas Young by more than 50 years. Euler, Daniel and Johann II as well as Nicholas II were all contemporaries as students of Johann I in Basel.

More Relations

Over the years, there were many more Bernoullis who followed in the family tradition. Some of these include:

Johann II Bernoulli (1710–1790; also known as Jean), son of Johann, mathematician and physicist

Johann III Bernoulli (1744–1807; also known as Jean), son of Johann II, astronomer, geographer and mathematician

Jacob II Bernoulli (1759–1789; also known as Jacques), son of Johann II, physicist and mathematician

Johann Jakob Bernoulli (1831–1913), art historian and archaeologist; noted for his Römische Ikonographie (1882 onwards) on Roman Imperial portraits

Ludwig Bernoully (1873 – 1928), German architect in Frankfurt

Hans Bernoulli (1876–1959), architect and designer of the Bernoullihäuser in Zurich and Grenchen SO

Elisabeth Bernoulli (1873-1935), suffragette and campaigner against alcoholism.

Notable marriages to the Bernoulli family include the Curies (Pierre Curie was a direct descendant to Johann I) as well as the German author Hermann Hesse (married to a direct descendant of Johann I).

References

[1] Calinger, Ronald S.. Leonhard Euler : Mathematical Genius in the Enlightenment, Princeton University Press (2015).

[2] Euler L and Truesdell C. Leonhardi Euleri Opera Omnia. Series secunda: Opera mechanica et astronomica XI/2. The rational mechanics of flexible or elastic bodies 1638-1788. (Zürich: Orell Füssli, 1960).

[3] D Speiser, Daniel Bernoulli (1700-1782), Helvetica Physica Acta 55 (1982), 504-523.

[4] Leibniz GW. Briefwechsel zwischen Leibniz, Jacob Bernoulli, Johann Bernoulli und Nicolaus Bernoulli. (Hildesheim: Olms, 1971).

[5] Hakfoort C. Optics in the age of Euler : conceptions of the nature of light, 1700-1795. (Cambridge: Cambridge University Press, 1995).

June 29, 2020January 5, 2021 by David D. Nolte

Johann Bernoulli’s Brachistochrone

Johann Bernoulli was an acknowledged genius–and he acknowledged it of himself. Some flavor of his character can be seen in his opening lines of one of the most famous challenges in the history of mathematics—the statement of the Brachistrochrone Challenge.

“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.”

Of course, he meant his own fame, because he thought he already had a solution to the problem he posed to the mathematical community of the day.

The Problem of Fastest Descent

The problem posed by Johann Bernoulli was the brachistochrone (Gk: brachis + chronos) or the path of fastest descent.

Galileo had attempted to tackle this problem in his Two New Sciences and had concluded, based on geometric arguments, that the solution was a circular path. Yet he hedged—he confessed that he had reservations about this conclusion and suggested that a “higher mathematics” would possibly find a better solution. In fact he was right.

Fig. 1 Galileo considered a mass falling along different chords of a circle starting at A. He proved that the path along ABG was quicker than along AG, and ABCG was quicker than ABG, and ABCDG was quicker than ABCG, etc. In this way he showed that the path along the circular arc was quicker than any set of chords. From this he inferred that the circle was the path of quickest descent—but he held out reservations, and rightly so.

In 1659, when Christiaan Huygens was immersed in the physics of pendula and time keeping, he was possibly the first mathematician to recognize that a perfect harmonic oscillator, one whose restoring force was linear in the displacement of the oscillator, would produce the perfect time piece. Unfortunately, the pendulum, proposed by Galileo, was the simplest oscillator to construct, but Huygens already knew that it was not a perfect harmonic oscillator. The period of oscillation became smaller when the amplitude of the oscillation became larger. In order to “fix” the pendulum, he searched for a curve of equal time, called the tautochrone, that would allow all amplitudes of the pendulum to have the same period. He found the solution and recognized it to be a cycloid arc.

On a cycloid whose axis is erected on the perpendicular and whose vertex is located at the bottom, the times of descent, in which a body arrives at the lowest point at the vertex after having departed from any point on the cycloid, are equal to each other…

His derivation filled 16 pages with geometric arguments, which was not a very efficient way to derive the thing.

Almost thirty years later, during the infancy of the infinitesimal calculus, the tautochrone was held up as a master example of an “optimal” solution whose derivation should yield to the much more powerful and elegant methods of the calculus. Jakob Bernoulli, Johann’s brother, succeeded in deriving the tautochrone in 1690 using the calculus, using the term “integral” for the first time in print, but it was not at first clear what other problems could yield in a similar way.

Then, in 1696, Johann Bernoulli posed the brachistrochrone problem in the pages of Acta Eruditorum.

Fig. 2 The shortest-time route from A to B, relying only on gravity, is the cycloid, compared to the parabola, circle and linear paths. Johann and Jakob Bernoulli, brothers, competed to find the best solution.

Acta Eruditorum

The Acta Eruditorum was the German answer to the Proceedings of the Royal Society of London. It began publishing in Leipzig in 1682 under the editor Otto Mencke. Although Mencke was the originator, launching and supporting the journal became the obsession of Gottfried Lebiniz, who felt he was a hostage in the backwaters of Hanover Germany but who yearned for a place on the world stage (i.e. Paris or London). By launching the continental publication, the Continental scientists had a freer voice without needing to please the gate keepers at the Royal Society. And by launching a German journal, it gave German scientists like Leibniz (and the Bernoullis and Euler, and von Tschirnhaus among others) a freer voice without censor by the Journal des Savants of Paris.

The Acta Eruditorum was almost a vanity press for Leibniz. He published 13 papers in the journal in its first 4 years of activity starting in 1682. In return, when Leibniz became embroiled in the priority dispute with Newton over the invention of the calculus, the Acta provided loyal support for Leibniz’ side just as the Proceedings of the Royal Society gave loyal support to Newton. In fact, the trigger that launched the nasty battle with Newton was a review that Leibniz wrote for the Acta in 17?? [Ref] in which he presented himself as the primary inventor of the calculus. When he failed to give due credit, not only to Newton, but also to lesser contributors, they fought back by claiming that Leibniz had stolen the idea from Newton. Although a kangaroo court by the Royal Society found in favor of Newton, posterity gives most of the credit for the development and dissemination of the calculus to Leibniz. Where Newton guarded his advances jealously and would not explain his approach, Leibniz freely published his methods for all to see and to learn and to try out for themselves. In this open process, the Acta was the primary medium of communication and gets the credit for being the conduit by which the calculus was presented to the world.

Although the Acta Eruditorum only operated for 100 years, it stands out as the most important publication for the development of the calculus. Leibnitz published in the Acta a progressive set of papers that outlined his method for the calculus. More importantly, his papers elicited responses from other mathematicians, most notably Johann Bernoulli and von Tschirnhaus and L’Hopital, who helped to refine the methods and advance the art. The Acta became a collaborative space for this team of mathematicians as they fine-tuned the methods as well as the notations for the calculus, most of which stand to this day. In contrast, Newton’s notations have all but faded, save the simple “dot” notation over variables to denote them as time derivatives (his fluxions). Therefore, for most of continental Europe, the Acta Eruditorum was the place to publish, and it was here that Johann Bernoulli published his famous challenge of the brachistochrone.

The Competition

Johann suggested the problem in the June 1696 Acta Eruditorum

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

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

The competition was originally proposed for 6 months, but it was then extended to a year and a half. Johann published his results about a year later, but not without controversy. Johann had known that his brother Jakob was also working on the problem, but he incorrectly thought that Jakob was convinced that Galileo had been right, so Johann described his approach to Jakob thinking he had little to fear in the competition. Johann didn’t know that Jakob had already taken an approach similar to Johann’s, and even more importantly, Jakob had done the math correctly. When Jakob showed Johann his mistake, he also ill-advisedly showed him the correct derivation. Johann sent off a manuscript to Acta with the correct derivation that he had learned from Jakob.

Within the year and a half there were 4 additional solutions—all correct—using different approaches. One of the most famous responses was by Newton (who as usual did not give up his method) but who is reported to have solved the problem in a day. Others who contributed solutions were Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus, and Guillaume de l’Hôpital’s. Of course, Jakob sent in his own solution, although it overlapped with the one Johann had already published.

The Solution of Jakob and Johann Bernoulli

The stroke of genius of Jakob and Johann Bernoulli, accomplished in 1697 only about 20 years after the invention of the calculus, was to recognize an amazing analogy between mechanics and light. Their insight foreshadowed Lagrange by a hundred years and William Rowan Hamilton by a hundred and fifty. They did this by recognizing that the path of a light beam, just like the trajectory of a particle, conserves certain properties. In the case of Fermat’s principle, a light ray refracts to take the path of least time between two points. The insight of the Bernoulli’s is that a mechanical particle would behave in exactly the same way. Therefore, the brachistrochrone can be obtained by considering the path that a light beam would take if the light ray were propagating through a medium with non-uniform refractive index to that the speed of light varies with height y as

Fermat’s principle of least time, which is consistent with Snell’s Law at interfaces, imposes the constraint on the path

This equation for a light ray propagating through a non-uniform medium would later become known as the Eikonal Equation. The conserved quantity along this path is the value 1/v_m. Rewriting the Eikonal equation as

it can be solved for the differential equation

which those in the know (as certainly the Bernoullis were) would know is the equation of a cycloid. If the sliding bead is on a wire shaped like a cycloid, there must be a lowest point for which the speed is a maximum. For the cycloid curve of diameter D, this is

Therefore, the equation for the brachistochrone is

which is the differential equation for an inverted cycloid of diameter D.

Fig. 4 A light ray enters vertically on a medium whose refractive index varies as the square-root of depth. The path of least time for the light ray to travel through the material is a cycloid—the same as for a massive particle traveling from point A to point B.

Calculus of Variations

Variational calculus had not quite been invented in time to solve the Brachistochrone, although the brachistochrone challenge helped motivate its eventual development by Euler and Lagrange later in the eighteenth century. Nonetheless, it is helpful to see the variational solution, which is the way we would solve this problem if it were a Lagrangian problem in advanced classical mechanics.

First, the total time taken by the sliding bead is defined as