Johann Bernoulli

How to use this course

Johann Bernoulli did more than anyone except Gottfried Wilhelm Leibniz himself to turn the calculus from a few short papers into a working method. He solved the catenary, posed the brachistochrone, wrote the first lectures on the differential and integral calculus, built the exponential calculus, and helped found the calculus of variations. He also taught Leonhard Euler. He was quarrelsome and jealous, fought publicly with his brother and his son, and has been accused, with reason, of taking credit that was not his. This course presents the mathematics and the man together.

Each technical lesson starts with a paragraph for the general reader, then gives precise statements in modern notation, derivations, and exercises with folded solutions. Bernoulli wrote in Leibniz’s notation, so his formulas look much more modern than those of his English contemporaries.

Timeline

A Basel family, 1667–1690

Johann Bernoulli was born in Basel on 6 August 1667 (27 July in the Julian calendar then still used there). His family were Protestant refugees from Antwerp who had settled in Basel and prospered in the spice trade. His father, Nicolaus, was a merchant and town official who intended his sons for useful careers. Johann was the tenth child. His brother Jacob, twelve years older, had been told to study theology.

Johann was first put to work in the family business, briefly, in 1682. He did not take to it. In 1683 he entered the University of Basel. His father wanted him to become a physician, and medicine remained his official subject for the next eleven years. His real education took place elsewhere. Jacob had defied their father and taught himself mathematics, and in 1687 he became professor of mathematics at Basel. Johann began to study with him.

Johann’s first publications were medical. A dissertation on fermentation appeared in 1690. His doctoral dissertation of 1694, on the motion of muscles, treated muscle contraction mathematically, with the new calculus. Leibniz reviewed it.

Two brothers and Leibniz’s calculus

In 1684 Leibniz published a six-page paper in Acta Eruditorum, the Leipzig journal, setting out the rules of his differential calculus. Two years later he published the integral calculus. The papers were terse and full of misprints. Few readers understood them. Jacob and Johann did. Working together through the late 1680s, the brothers mastered Leibniz’s methods and began to extend them.

At first the brothers were partners. They worked on caustics, the envelopes of reflected and refracted rays, in 1692 and 1693. Yet they never published a joint paper, and the partnership turned to rivalry. Johann resented being treated as a pupil. Jacob resented his younger brother’s boasting. In 1691 Jacob posed the problem of the hanging chain in Acta Eruditorum. Johann solved it, and Jacob did not, an outcome Johann recalled with pleasure for the rest of his life (Lesson 04).

By the mid-1690s the quarrel was conducted in public, in the pages of the same journals, with challenge problems and insults. It shaped Johann’s best work, because each brother set problems designed to defeat the other.

Paris and the Marquis, 1691–1695

In 1691 Johann lectured on the calculus in Geneva and then went to Paris. There he joined the circle of Nicolas Malebranche, where he was one of the very few people in France who understood Leibniz’s methods. He met Pierre Varignon in 1692, beginning a lifelong correspondence. Most importantly, he met Guillaume François Antoine, Marquis de l’Hôpital, a capable mathematician who wanted to learn the new analysis.

Bernoulli gave l’Hôpital lessons in Paris and at his country estate at Oucques, and continued them by letter after returning to Basel. From 1694 the arrangement was formal: l’Hôpital paid him a regular sum, and in return Bernoulli sent him his mathematical discoveries and agreed not to share them with others. One letter, dated 22 July 1694, contained the rule for evaluating \(0/0\) (Lesson 05).

In 1696 l’Hôpital published Analyse des infiniment petits pour l’intelligence des lignes courbes, the first textbook of the differential calculus. Its preface thanked Leibniz and the Bernoullis, but the book drew mostly on Johann’s lessons. Bernoulli said nothing in public until after l’Hôpital’s death in 1704, when he claimed much of the book as his. Many readers dismissed the claim as typical Bernoulli boasting. In 1922 a copy of his differential calculus lectures was found in Basel, and their close agreement with the Analyse showed that he had largely been right.

Meanwhile he married Dorothea Falkner, daughter of a Basel family, in 1694. In 1695, on the recommendation of Christiaan Huygens, he was offered the chair of mathematics at Groningen in the Netherlands. He arrived there with his wife and infant son in October 1695.

The catenary

For the general reader. A chain hanging between two hooks looks like a parabola, and Galileo suggested that it was. It is not. In 1691 Jacob Bernoulli challenged mathematicians to find the true curve. Johann, Leibniz and Huygens all solved it. For Johann, just 24, it was the first major success, and a demonstration of what the new calculus could do.

The equation

Let a uniform chain of weight \(w\) per unit length hang in a vertical plane, with its lowest point on the \(y\)-axis. Consider the piece from the lowest point to a point \(P\) at arc length \(s\). Three forces hold it in equilibrium: the horizontal tension \(H\) at the bottom, the weight \(ws\), and the tension at \(P\), which acts along the tangent. Balancing horizontal and vertical components, the tangent at \(P\) satisfies \(\tan\theta=ws/H\). Writing \(a=H/w\),

Derivation. Put \(p=dy/dx\). Then \(dp/\sqrt{1+p^2}=dx/a\), so \(\operatorname{arsinh}p=x/a\) (using \(p=0\) at \(x=0\)), and \(p=\sinh(x/a)\). Integrating again gives \(y=a\cosh(x/a)\). Bernoulli had no hyperbolic functions. He constructed the curve geometrically from the area under a hyperbola, that is, from logarithms, which is the same thing. For the tension, \(T=H\sec\theta=H\sqrt{1+p^2}=H\cosh(x/a)=wy\).

The rule for 0/0

For the general reader. What is the value of a fraction when both top and bottom become zero? Bernoulli’s answer, sent to l’Hôpital in July 1694, is to compare how fast each vanishes, using their derivatives. The rule bears l’Hôpital’s name because it first appeared in his 1696 textbook.

The original statement

The Analyse considers a curve whose ordinate is \(y=f(x)/g(x)\) with \(f(a)=g(a)=0\). At \(x=a\) both numerator and denominator are “zero”, but their differentials \(df\) and \(dg\) are not. Replacing \(x=a\) by the infinitely near point \(a+dx\), the ordinate becomes \(\frac{f(a)+df}{g(a)+dg}=\frac{df}{dg}\). The book applies this to

finding \(y=\tfrac{16}{9}a\).

Proof. Extend \(f,g\) by \(f(a)=g(a)=0\). For \(x\neq a\), Cauchy’s mean value theorem gives \(\xi\) strictly between \(a\) and \(x\) with \(\big(f(x)-f(a)\big)g'(\xi)=\big(g(x)-g(a)\big)f'(\xi)\), that is \(f(x)/g(x)=f'(\xi)/g'(\xi)\). (Note \(g(x)\neq0\), else Rolle’s theorem would give a zero of \(g'\).) As \(x\to a\), \(\xi\to a\), and the right side tends to \(L\).

Series and the exponential calculus

For the general reader. Leibniz’s rules handle powers like \(x^3\). Bernoulli extended them to expressions where the variable sits in the exponent, such as \(x^x\). He also found a general series for integrals by repeated integration by parts, and an astonishing identity relating an integral of \(x^{-x}\) to a sum of \(n^{-n}\).

Bernoulli’s series (1694)

Derivation. Integrate by parts repeatedly: \(\int_0^x y(t)\,dt=\big[(t-x)y(t)\big]_0^x-\int_0^x(t-x)y'(t)\,dt=xy(0)+\cdots\). Bernoulli arranged the steps so the derivatives are taken at the upper limit, which amounts to expanding around \(x\): \(y(t)=\sum_k y^{(k)}(x)(t-x)^k/k!\), integrated term by term. The series is a close relative of the one Brook Taylor published in 1715, and Bernoulli later claimed that Taylor’s series was a consequence of his.

The exponential calculus (1697)

Bernoulli called \(x^x\), \(a^x\) and similar expressions “percurrent” and showed how to differentiate them by taking logarithms: if \(y=x^x\), then \(\ln y=x\ln x\), so \(dy=x^x(1+\ln x)\,dx\).

Proof. Expand \(x^{-x}=e^{-x\ln x}=\sum_{k\ge0}\frac{(-x\ln x)^k}{k!}\), which converges uniformly on \([0,1]\) since \(|x\ln x|\le 1/e\). Substituting \(x=e^{-u}\) gives \(\int_0^1 x^k(-\ln x)^k\,dx=\int_0^\infty u^ke^{-(k+1)u}\,du=\frac{k!}{(k+1)^{k+1}}\). Summing, \(\int_0^1x^{-x}dx=\sum_{k\ge0}(k+1)^{-(k+1)}\). The second identity is the same with alternating signs.

The brachistochrone

For the general reader. Which slide gets a ball from one point to a lower one fastest? Not the straight ramp. In June 1696 Bernoulli challenged the mathematicians of Europe to find the curve. His own solution treated the falling ball as a ray of light passing through layers of glass, each faster than the last. The answer is a cycloid, the curve traced by a point on a rolling wheel.

The challenge

Bernoulli posed the problem in Acta Eruditorum in June 1696. At Leibniz’s suggestion he extended the six-month deadline to Easter 1697, with a printed announcement dated 1 January 1697, which he sent to scholars across Europe. Besides his own, five solutions came in: from Jacob Bernoulli, Leibniz, Ehrenfried Walther von Tschirnhaus, l’Hôpital, and an anonymous solution from England. Isaac Newton had received the problem on 29 January 1697 and solved it overnight. Bernoulli recognized the author, he said, as one knows the lion by its claw. The solutions appeared in the Acta in May 1697.

Bernoulli’s solution

Let the curve start at the origin with \(y\) measured downward. A body falling from rest has speed \(v=\sqrt{2gy}\) by energy conservation. Fermat’s principle says light travels the path of least time, and Snell’s law expresses that principle: at each interface \(\sin\theta/v\) is unchanged, where \(\theta\) is the angle from the vertical. Bernoulli sliced the plane into thin horizontal layers, each with speed \(\sqrt{2gy}\), and concluded that along the fastest path

Derivation. Since \(\sin\theta=dx/ds=1/\sqrt{1+y'^2}\), the constant condition reads \(y(1+y'^2)=D\) for a constant \(D\). Solving, \(dx=\sqrt{y/(D-y)}\,dy\). Substitute \(y=D\sin^2(\varphi/2)=\tfrac D2(1-\cos\varphi)\). Then \(dy=D\sin(\varphi/2)\cos(\varphi/2)\,d\varphi\) and \(\sqrt{y/(D-y)}=\tan(\varphi/2)\), so \(dx=D\sin^2(\varphi/2)\,d\varphi=\tfrac D2(1-\cos\varphi)\,d\varphi\). Integrating, \(x=\tfrac D2(\varphi-\sin\varphi)\): a cycloid with \(r=D/2\).

Integration and differential equations

For the general reader. Differentiating is mechanical; integrating is an art. Bernoulli’s lectures on the integral calculus, written for l’Hôpital in 1691–92, were the first systematic attempt to teach it. He and Leibniz showed how to integrate every rational function by splitting it into simple fractions, and Johann developed methods for whole families of differential equations.

Rational functions

The proof factors \(Q\) into real linear and quadratic factors and decomposes into partial fractions. Bernoulli relied on the assertion, then unproved, that every real polynomial so factors; Leibniz doubted it, citing \(x^4+a^4\), until it was shown that \(x^4+a^4=(x^2+\sqrt2ax+a^2)(x^2-\sqrt2ax+a^2)\). The general fact is the fundamental theorem of algebra, proved only in the nineteenth century.

Bernoulli’s equation

In 1695 Jacob Bernoulli proposed the equation below. Leibniz and Johann solved it, Johann by writing the unknown as a product of two functions.

Logarithms of negative numbers

In correspondence of 1712–13 Leibniz argued that \(\log(-1)\) is imaginary. Bernoulli argued that \(\log(-x)=\log x\), since \(d(-x)/(-x)=dx/x\). Both had part of the truth. Euler, Bernoulli’s student, resolved the question in the late 1740s: the logarithm of \(-1\) has infinitely many values \((2k+1)\pi i\).

Isoperimetric problems and geodesics

For the general reader. The brachistochrone asks for the best curve, not the best number. Problems of that kind make up the calculus of variations. Jacob answered his brother’s challenge with a harder one, about curves of fixed length, and the quarrel that followed produced methods that Euler and Lagrange turned into a general theory.

The general equation

For a functional \(J[y]=\int_a^b F(x,y,y')\,dx\) with fixed end values, varying \(y\) to \(y+\varepsilon\eta\) with \(\eta(a)=\eta(b)=0\) and integrating by parts gives the condition below. Euler derived it in 1744, building on the Bernoullis’ special methods; Lagrange gave the modern variational derivation in 1755.

For the brachistochrone, \(F=\sqrt{(1+y'^2)/y}\), and the first integral gives \(y(1+y'^2)=\text{const}\), Bernoulli’s equation from Lesson 07.

The isoperimetric problem

Jacob’s 1697 problem asked, among curves of given length, for those maximizing an area of a certain type. Johann’s first answer was flawed, and Jacob pointed this out. Johann’s mature solution of 1718 introduced ideas that Euler later systematized. In modern form, to make \(\int F\) stationary subject to \(\int G=\text{const}\), one applies the Euler–Lagrange equation to \(F-\lambda G\) for a constant multiplier \(\lambda\).

Geodesics

In 1697 Johann posed the problem of the shortest path between two points on a convex surface. He showed in 1698 that along such a path the osculating plane always contains the normal to the surface. Equivalently, the acceleration of a point moving along the geodesic at unit speed is normal to the surface. Euler, in one of his early papers, derived the differential equation of geodesics.

Virtual work and living force

For the general reader. When is a machine in balance? Bernoulli gave a single rule: imagine every possible small motion consistent with the machine’s constraints; it is balanced if the forces do no net work in any of them. He also defended Leibniz’s “living force”, \(mv^2\), the ancestor of kinetic energy, in a long debate about what is conserved in collisions.

Virtual work (1717)

In a letter to Varignon dated 26 January 1717, Bernoulli stated what he called the principle of virtual velocities. Varignon published it in his Nouvelle mécanique (1725). Lagrange made it the starting point of the Mécanique analytique (1788).

Living force

René Descartes had held that the quantity of motion, \(mv\), is conserved; Leibniz argued for \(mv^2\), which he called vis viva. Bernoulli took Leibniz’s side in his Discours sur les loix de la communication du mouvement, written for the Paris Academy prize competition of 1724 and published in 1727. He treated vis viva as conserved, with the apparent losses in inelastic collisions absorbed by the compression of the bodies. The dispute was largely about words. In modern terms momentum \(mv\) and energy \(\tfrac12mv^2\) are both conserved in elastic collisions, and only momentum in inelastic ones, the lost kinetic energy going into heat and deformation.

Groningen and Basel, 1695–1748

Bernoulli spent ten years at Groningen. He lectured on mathematics and experimental physics and became involved in theological quarrels with the Calvinist clergy, who suspected his views. His son Daniel was born there in 1700. His nephew Nicolaus Bernoulli, later known as Nicolaus I, came to study with him. Through these years the public conflict with Jacob over the isoperimetric problem continued in print.

In 1705 his wife’s family pressed him to return to Basel, and he set out. On the way he learned that Jacob had died in August. He was appointed to his brother’s chair of mathematics at Basel and held it for the remaining 42 years of his life, declining offers from Utrecht, Leiden and Groningen. He was elected a Fellow of the Royal Society in 1712 and was a member of the academies of Paris, Berlin, St Petersburg and Bologna.

From 1713 Bernoulli was Leibniz’s chief champion in the priority dispute with Newton’s followers. He set problems meant to show the superiority of Leibniz’s methods and wrote anonymous attacks. He also defended Descartes’ theory of vortices against Newtonian gravitation, which slowed the acceptance of Newton’s physics on the Continent.

In the 1720s a young student named Leonhard Euler came to the university. Bernoulli declined to give him private lessons but invited him to bring his difficulties on Sunday afternoons. He soon recognized Euler’s talent and persuaded Euler’s father, an old acquaintance, to let his son study mathematics rather than theology. Bernoulli’s sons Nicolaus II and Daniel went to the new St Petersburg Academy in 1725; when Nicolaus died there in 1726, Euler was invited in his place.

Quarrels, students and legacy

Bernoulli’s quarrels outlived Jacob. In 1734 he and his son Daniel shared the Grand Prize of the Paris Academy. Johann was so angry at being rated his son’s equal that he barred Daniel from his house. Worse followed. In 1738 Daniel published Hydrodynamica. Johann’s Hydraulica, which drew heavily on his son’s work, bore the date 1732, making it appear to come first. Historians regard the dating as a deliberate attempt to take credit for Daniel’s ideas.

In 1742 his collected works, Opera omnia, appeared in four volumes, including the integral calculus lectures written for l’Hôpital half a century earlier. He died in Basel on 1 January 1748, aged 80. His epitaph calls him the Archimedes of his age.

Legacy

• Teaching: through l’Hôpital’s Analyse, his own lectures and his students, especially Euler, Daniel Bernoulli, Pierre Louis Maupertuis and Johann Samuel König, he spread Leibniz’s calculus across Europe.
• Analysis: the catenary, the rule for indeterminate forms, the exponential calculus, partial fractions, and methods for differential equations.
• Variational calculus: the brachistochrone, the isoperimetric quarrel and geodesics led directly to Euler’s and Lagrange’s general theory.
• Mechanics: the principle of virtual work, the basis of analytical mechanics.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Johann Bernoulli, MacTutor History of Mathematics.
2. Johann Bernoulli, Wikipedia.
3. J. J. O’Connor and E. F. Robertson, Daniel Bernoulli and Leonhard Euler, MacTutor.
4. O. Spiess (ed.), Der Briefwechsel von Johann Bernoulli, vol. 1, Birkhäuser, 1955.

Mathematics

1. Johann Bernoulli, Opera omnia, 4 vols., Lausanne and Geneva, 1742.
2. G. F. A. de l’Hôpital, Analyse des infiniment petits pour l’intelligence des lignes courbes, Paris, 1696.
3. Brachistochrone curve, Wikipedia (history and Bernoulli’s optical method).
4. L’Hôpital’s rule, Wikipedia (history of the 1694 letter).
5. Sophomore’s dream, Wikipedia.
6. H. H. Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century, Springer, 1980.
7. C. Truesdell, The rational mechanics of flexible or elastic bodies 1638–1788, in Euler, Opera omnia ser. II vol. 11, 1960.