Johann Bernoulli was
born in Basel, on 7th August ** 1667**,
the tenth child (of eleven) of Nicolaus Bernoulli and Margaretha (neé Schönawer), twelve years younger than his brother Jakob.

His parents had plans for Johann to move into the family spice business. In ** 1682**, at the age of 15 years old, he worked in the spice trade for a year (in Neuchatel) but by all accounts did not take to the work at all.

Despite parental reluctance, in ** 1683**, Johann entered Basel University to study medicine. Paradoxically, despite the family's fame for mathematics, many members of the family actually studied medicine.

His brother Jakob was lecturing on experimental physics when Johann entered the university and soon he was engaged in studying mathematics with Jakob - notably Leibniz's papers on calculus.

In December **1685**, magister from Philosopht Professor, Nicolaus Eglinger.

Johann's first publication, in **1690**, was not a mathematical thesis - it was on the process of fermentation, and published in connection with the otaining of his licence in Medicine.

A few months later, first published in mathematics in *Acta Eruditorum*

### Geneva and Paris

In **1691** Johann went to Geneva where he lectured on the differential calculus, to J.C. Fatio-de-Duillier, the brother of Jakob's friend.

From there he went to Paris and became acquainted with mathematicians of Malebranche's circle, a focus of French mathematics at that time, and Johann was welcomed as a representative of the new differential calculus. It was there that he met Guillaume de L'Hôpital.

De L'Hôpital asked Johann to teach him the new calculus of Leibniz, which he did both in Paris and at de L'Hôpital's country house at Oucques, west of Orleans. Bernoulli received a generous income for these lessons (about half of the salary received by a professor) and after he returned to Basel, he still continued by correspondence. This instruction enabled L'Hopital, in **1696**, to publish the first calculus book *Analyse des infiniment petits pour l'intelligence des lignes courbes *. Unfortunately, the work did not acknowledge Bernoulli's contributions as such. The preface of the book contains only the statement:-

And then I am obliged to the gentlemen Bernoulli for their many bright ideas; particularly to the younger Mr Bernoulli who is now a professor in Groningen.

L'Hôpital's rule is in the book, and it received this name despite being a finding of Johann's. It was first attributed to L'Hôpital by Saurin during an argument with Rolle. In reality, proof that the work was due to Johann was not obtained until **1922** when a copy of Johann's course made by his nephew Nicolaus Bernoulli was found in Basel. Bernoulli's course is virtually identical with de l'Hôpital's book but de l'Hôpital had corrected a number of errors such as Bernoulli's mistaken belief that the integral of 1/x is finite (l'Hôpital did have some major ability in maths in his own right).

After L'Hôpital's death in **1704**, Johann protested strongly that the book contained his ideas (he seems to have been bound by his agreement with L'Hôpital from speaking out earlier). Nevertheless this claim was not universally believed until the 1922 discovery of a copy of Johann's courses that had been made by his nephew, Nicolaus.

In **1692**, he met Varignon and this resulted in a strong friendship and Varignon learned much about applications of the calculus from Johann Bernoulli over the many years which they corresponded.

Johann Bernoulli also began a correspondence with Leibniz which was to prove very fruitful. In fact this turned out to be the most major correspondence which Leibniz carried out.

### Catenary

In **1691**, Johann solved the problem of the catenary (the shape of a chain hanging in equilibrium) which had been posed by his brother in the same year (it had been solved by Leibniz and also by Huygens, the latter being the first person to use the name). He was able to form a differential equation and it was his first important mathematical result produced independently of his brother, although it used ideas that Jakob had given when he posed the problem.

The catenary is the locus of the focus of a parabola rolling along a straight line.

The catenary is the evolute of the tractrix.

It is the locus of the mid-point of the vertical line segment between the curves e^{x} and e^{-x}. Therefore it has the cartesian equation

y = a cosh(x/a)

Euler showed in **1744** that a catenary revolved about its asymptote generates the only minimal surface of revolution.

### Velaria

The velaria was the name given to the shape of sail in wind. In **1691**, on the way to Paris, he found it was a catenary. The solution was achieved by solving a differential equation formulated by Jakob. Jakob himself reached the same conclusion at a later date, using a different method.

### Johann and Jakob

On his return to Basel, he took on a job as an engineer with the City Council. At this stage Johann and Jakob were still learning much from each other in a reasonably friendly rivalry (possibly). For example they worked together on caustic curves during **1692-93** although they did not publish the work jointly. Even at this stage the rivalry was too severe to allow joint publications and they would never publish joint work at any time despite working on similar topics.

This was a period of considerable mathematical achievement for Johann Bernoulli. Although he was working on his doctoral dissertation in medicine he was producing numerous papers on mathematical topics which he was publishing and also important results which were contained in his correspondence.

Johann's doctoral dissertation was submitted in **1694**, being an application of mathematics to medicine relating to muscular movement.

Also in **1694**, Bernoulli produced his 'universal series' which is similar to the Taylor Series.
This series is still known as the Bernoulli Series in continental Europe. Taylor announced his series in **1715**, causing Johann to seethe with suspicions of plagiarism.

In **1694**, he considered

y = x^{x}

In 1694, became Doctor of Medicine with a dissertation applying calculus to muscle contractions.

In the same year, he married Dorothea Falkner, on accepting a job as a land surveyor.

### Groningen

In **1695** he was offered chairs in Groningen and Halle, the former via the agency of Huygens. He had already decided on Groningen when the Halle offer arrived (via the agency of Leibniz). Seemingly he would have preferrd to go to halle but was already committed to Groningen where he eventually stayed for 10 years. Although appointed to the Chair of Mathematics, his letter of appointment mentions his medical skills and offered him the chance to practice medicine while in Groningen.

Johann had married Dorothea Falkner and their first child (Nicolaus(II)) was only seven months old when they departed for Groningen. Setting out on 1 September 1695 they had to cross a war zone, among other things. They arrived on 22 October.

These ten years were filled with several difficulties. For one thing, he was involved in a number of religious disputes and accused of being a follower of Spinoza. Also, his second child, a daughter, was born in **1697** and only lived for six weeks, and he suffered so severe an illness that he was reported to have died.

In one dispute he was accused of denying the resurrection of the body, a charge based on medical opinions he held. He had to defend himself before a public hearing.

A second dispute in **1702** involved acccusations from a student at the university, Petrus Venhuysen, who published a pamphlet accusing him of following Descartes' philosophy and of opposing the Calvinist faith and depriving believers of their comfort in Christ's passion. As part of a lengthy response, Bernoulli replied

... I would not have minded so much if [Venhuysen] had not been one of the worst students, an utter ignoramus, not known, respected, or believed by any man of learning, and he is certainly not in a position to blacken an honest man's name, let alone a professor known throughout the learned world...

He attracted some criticism for introducing physics experiments into his teaching, which were seemingly anathema to some Cartesians and Calvinists alike.

An important discovery was a fluorescent light emitted by mercury due to friction inside barometers.

The 18th Century saw trigonometric functions of a complex variable being studied. Johann Bernoulli found the relation between sin^{-1}z and log z in **1702**.

### Brachistochrone

Johann competed with his brother in a couple of mathematical problems which seem to have become embroiled in their bitter personal battle (enriching mathematics along the way, according to some sources). Although knowing the solution himself (a cycloid), Johann proposed the problem of the brachistochrone in June **1696** and challenged others to solve it.

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.

He stated the problem in *Acta Eruditorum* (a journal published in Leipzig) and four other solutions were obtained, from Newton, Jakob Bernoulli, Leibniz and de L'Hôpital.

Johann's solution to the brachistochrone problem was less satisfactory than that of Jakob but when Johann returned to the problem in 1718 having read a work by Taylor, he produced an elegant solution which was to form a foundation for the Calculus of Variations.

(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)

Not to be outdone by his brother, Jakob issued his own challenge. Returning to an poser considered by Galileo involving 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

In **1702** Jakob then proposed the isoperimetric problem (minimising the area enclosed by a curve) which again led to bitter disputes between the two brothers, which Varignon also became involved in. Returning to this problem in 1718, Johann found greater insights into the problem, leading on to the calculus of variations.

An offer arived from Leiden University, where the aged Professor Volder had agreed to stand down in his favor.

In **1703**, they strted to decide on a return to Basel, where Johann's father-in-law was pining for his daughter and grandchildren and apparently did not have long to live. After an illness delayed things, they set off in **1705**.

Originally Johann had intended to fill the vacant Chair of Greek at Basel University but there is a story that he intended to accept a good offer from Utrecht University after a short stay in Basel, without giving them a full reassurance (see the story about Utrecht later on)

The family were accompanied by Nicolaus(I) Bernoulli, his nephew, who had been studying mathematics in Groningen with his uncle. They left Groningen two days after Jakob's death (of tuberculosis) but obviously they were unaware of this at the time and only heard the news *en route*, in Amsterdam.

1697 Acta

### Basel

Before reaching Basel, however, Johann was tempted by an offer of a chair at the University of Utrecht. The head of the University of Utrecht was so keen to have Bernoulli that he pursued him and caught up with him in Frankfurt. Unfortunately, Johann was set on returning to Basel.

On his return to Basel, Johann was soon he was appointed to Jakob's chair of mathematics (for the next 42 years). Meanwhile, his father-in-law lived for a further three years in which he greatly enjoyed having his daughter and grandchildren back in Basel.

His work in Experimental Physics was constricted by inferior instrumnents in comparison with Utrecht.

There were other offers that Johann also turned down, such as Leiden (after the death of Professor Volder in 1709), a second offer from Utrecht, an offer from Padua in **1714** and a generous offer for him to return to Groningen in **1717**.

It would be fair to say that Johann Bernoulli began the study of the calculus of the exponential function in 1697 when he published Principia calculi exponentialium seu percurrentium. The work involves the calculation of various exponential series and many results are achieved with term by term integration.

1712-1713 had argued with Leibniz in favor of the existence of logs of complex numbers

Argued since

Then

log(-x) = log(x)

implying

log(-1) = log(1) = 0

In **1713** Johann became involved in the Newton-Leibniz controversy. He strongly supported Leibniz and added weight to the argument by showing the power of his calculus in solving certain problems which Newton had failed to solve with his methods. Although Bernoulli was essentially correct in his support of the superior calculus methods of Leibniz, he also supported Descartes' vortex theory over Newton's theory of gravitation and here he was certainly incorrect. His support in fact delayed acceptance of Newton's physics on the Continent.

Integration to Johann was viewed simply as the inverse operation to differentiation and with this approach he had great success in integrating differential equations. He summed series, and discovered addition theorems for trigonometric and hyperbolic functions using the differential equations they satisfy

On the negative side, he was an opponent of the ideas of Newton on mechanics and a supporter of the alternative theory offered by Descartes. However, he did work thru Newton's *Principia* and corrected many errors.

In **1715**, he and Hermann, among others, clarified the notion that a surface can be represented by an equation in three coordinates.

**1718** *Sur le manoeuvre des vaisseaux*

**1730** Paris, cause of form of orbits and changes of aphelia

**1734** inclination of planets to ecliptic, Daniel, Euler

Latterly, Johann became involved in some dispute with his own son Daniel. Daniel completed his most important work *Hydrodynamica* in **1734** and published it in **1738**. Very soon after, Johann published *Hydraulica*, although he dated it **1732** in an attempt to obtain priority over his own son.

He investigated

which he solved by introducing method of partial fractions (which was discovered independently by Leibniz).

His pupils included Leonhard Euler, Marquis de L’Hospital, Maupertuis, Gabriel Cramer, his son Johann II, Alexis Clairaut and Niklaus Blauner, Johann Samuel König.

He was succeeeded as Professor by Johann (II).

investigated series by using integration by parts

dispute continued 1727-31 between himself and Euler.

### Books

** **
*Opera Omnia*
Notes Moon 470/471, 557