Jacob Bernoulli

How to use this course

Jacob Bernoulli was the first mathematician in a family that produced eight of them over three generations. With his younger brother Johann he was among the first people to understand Gottfried Wilhelm Leibniz’s calculus, and he used it on a long list of problems: the isochrone, the hanging chain, the bent beam, the shape of a sail. He studied infinite series, found the numbers that now carry his name, and wrote the founding book of mathematical probability. That book, Ars Conjectandi, contains the first proof that observed frequencies settle down to probabilities. He did not live to publish it.

Each technical lesson starts with a paragraph for the general reader, then gives precise statements in modern notation, derivations and exercises with folded solutions. The lessons follow his work roughly in order: differential equations and curves (1690–1696), series (1689–1704), and probability, developed over twenty years and published in 1713. The companion course on Johann Bernoulli covers the catenary and the brachistochrone in detail.

Timeline

Theology and the stars, 1655–1676

Jacob Bernoulli was born in Basel on 6 January 1655, which was 27 December 1654 in the Julian calendar still used in the city. His grandfather, a druggist from Amsterdam, had become a citizen of Basel by marriage in 1622. The family were Protestants whose forebears had left the Spanish Netherlands, and in Basel they prospered in the spice and drug trade. His father, Nicolaus, ran the business, sat on the town council and served as a magistrate. His mother, Margaretha Schönauer, came from a family of bankers and councillors.

Nicolaus wanted his eldest son in the church. Jacob took a master’s degree in philosophy at the University of Basel in 1671 and a licentiate in theology in 1676. Alongside these studies, and against his father’s wishes, he taught himself mathematics and astronomy.

Twelve years younger, his brother Johann was born in 1667. The father had similar plans for him, first commerce and then medicine. Both sons would end as professors of mathematics, one after the other, in the same chair.

Travels, 1676–1682

After his theology degree Bernoulli went to Geneva in 1676 as a private tutor. In 1677 he began the Meditationes, a scientific notebook he kept for the rest of his life. Much of what we know about the slow development of his ideas on probability comes from it.

He then spent about two years in France. There he studied the natural philosophy of René Descartes with his followers, the circle led by Nicolas Malebranche. In 1681 and 1682 he made a second journey, to the Netherlands and England. In the Netherlands he met the mathematician Johannes Hudde, a former mayor of Amsterdam. In England he met Robert Boyle and Robert Hooke. The trip gave him a network of correspondents and a taste for experimental science.

The great comet of 1680 had excited all of Europe. Bernoulli’s first scientific publication, the Conamen novi systematis cometarum of 1682, proposed a theory of comets as bodies returning on regular paths. The theory was wrong. A tract on the weight of the ether followed in 1683.

The Basel chair and Leibniz’s calculus, 1683–1690

Back in Basel, Bernoulli began in 1683 to give lectures on the mechanics of solids and liquids, with experiments. In 1684 he married Judith Stupanus, the daughter of a wealthy pharmacist. They had a son, Nicolaus, who became a town councillor and master of the artists’ guild, and a daughter. Between 1684 and 1686 he published four disputations on logic, comparing logical and algebraic reasoning, and in 1685 his first paper on probability. In 1687 he became professor of mathematics at the University of Basel. He held the chair for the rest of his life.

In 1684 Leibniz had published the rules of the differential calculus in Acta Eruditorum, the Leipzig journal, and in 1686 the first account of the integral calculus. The papers were short and obscure. Bernoulli studied them with his brother Johann, who had entered the university in 1683 and was learning mathematics from him. By 1690 the brothers were among the very few people anywhere who could use the new methods.

The first sign of that skill was Jacob’s paper of May 1690 on the isochrone (Lesson 04). In the same paper he asked for the curve of a hanging chain. Johann solved it, with Leibniz and Christiaan Huygens; Jacob did not. Their solutions appeared in June 1691. It was the first of many contests between the brothers.

The isochrone and the word “integral”

For the general reader. In 1687 Leibniz asked for a curve along which a falling body descends at a steady rate, covering equal heights in equal times. Huygens and Leibniz found the answer but did not show how. In 1690 Jacob Bernoulli published a full analysis with the new calculus. The paper also gave the integral its name.

The problem

Measure \(y\) downward. A body released from rest at \(y=0\) falls freely to depth \(y_0\), where it has speed \(c=\sqrt{2gy_0}\), and then enters a frictionless curve. By energy conservation its speed at depth \(y\) is \(v=\sqrt{2gy}\). The isochrone condition is that the vertical speed stays equal to \(c\): \(v\,dy/ds=c\), where \(s\) is arc length.

Separation of variables

Squaring and using \(ds^2=dx^2+dy^2\),

Bernoulli wrote the equation in the same shape, with constants \(a,b\): \(dy\sqrt{b^2y-a^3}=dx\sqrt{a^3}\). He then observed that since the two differentials are equal, their integrals are equal. The Latin phrase he used, ergo et horum integralia aequantur (“therefore their integrals are also equal”), is the first printed use of “integral” in this sense. Leibniz had spoken of a calculus summatorius; he later adopted the Bernoullis’ term.

Derivation. Put \(u=2gy-c^2\), so \(du=2g\,dy\). Then \(c\,dx=\sqrt u\,du/(2g)\), and integrating from the cusp, \(cx=u^{3/2}/(3g)\). Squaring gives the result.

Compound interest and the number e

For the general reader. Lend one coin at 100% interest for a year. Paid once, you get 2. Paid in two halves, compounded, you get \(1.5^2=2.25\). Paid monthly, about 2.61. Does the amount grow without limit as the payments become more frequent? Bernoulli asked this in 1690 and showed that it does not: the answer lies between 2.5 and 3. The limit is the number now called \(e\).

The question appeared in his paper Quaestiones nonnullae de usuris (1690). Notebook entries suggest he had studied it in 1683. In modern terms:

Proof. By the binomial theorem,

Each product increases with \(n\), and \(a_{n+1}\) has one extra positive term, so \(a_n\) increases. Each term is at most \(1/k!\le 1/2^{k-1}\) for \(k\ge1\), so \(a_n<1+1+\tfrac12+\tfrac14+\cdots=3\). Hence the limit exists. Fixing \(m\) and letting \(n\to\infty\) in the first \(m\) terms gives \(e\ge\sum_{k\le m}1/k!\) for every \(m\); and \(a_n\le\sum_{k\le n}1/k!\). So \(e=\sum 1/k!\). In particular \(e>1+1+\tfrac12+\tfrac16>2.5\).

Curves: spiral, elastica, lemniscate

For the general reader. Bernoulli loved curves that reproduce themselves. The logarithmic spiral, which he called the marvellous spiral, looks the same at every scale, and many constructions applied to it give back the same spiral. He also asked what shape an elastic beam takes when it is bent, and in solving that problem he met a figure-eight curve now called the lemniscate of Bernoulli.

The spira mirabilis

Proof. (1) In polar coordinates \(\tan\psi=r/(dr/d\theta)=1/b\). (2) \(\lambda ae^{b\theta}=ae^{b(\theta+\ln\lambda/b)}\), so scaling by \(\lambda\) is rotation by \(\ln\lambda/b\). (3) In complex notation \(z(\theta)=ae^{(b+i)\theta}\), so \(z'=(b+i)z\) and \(z''=(b+i)^2z\). The centre of curvature of a plane curve is \(z+i\,z'\,|z'|^2/\operatorname{Im}(\bar z'z'')\). Here \(\bar z'z''=(1+b^2)(b+i)|z|^2\), so the centre is \(z+i(b+i)z=ibz\): the spiral scaled by \(b\) and turned by a quarter turn, which by (2) is congruent to the original. The radius of curvature is \(\rho=r\sqrt{1+b^2}\). Bernoulli also showed that the caustics and pedal curves of the spiral are again logarithmic spirals. He asked for one on his tombstone (Lesson 13).

The elastica

In 1691 Bernoulli announced, and in 1694 published, a treatment of the elastica, the shape of a thin elastic beam bent by forces at its ends. His physical hypothesis, in modern words, is that the curvature at each point is proportional to the bending moment there. Writing the curvature as \(\frac{d}{dx}\big(y'/\sqrt{1+y'^2}\big)\), the “rectangular” case, in which the bending moment grows linearly with \(x\), gives

The last form is Bernoulli’s equation of the rectangular elastica. The integral cannot be expressed through algebraic functions, logarithms and circular functions. It is an elliptic integral, among the first to appear in a physical problem.

The lemniscate

In 1694, working on the paracentric isochrone (Exercise 4.2), Bernoulli introduced the curve

It is the set of points whose distances to the foci \((\pm a/\sqrt2,0)\) have product \(a^2/2\). He showed that its arc length from the centre is \(s=\int_0^r a^2\,dr/\sqrt{a^4-r^4}\), the companion of the elastica integral. Giulio Carlo Fagnano, Leonhard Euler and Carl Friedrich Gauss studied this integral further, and Gauss’s work on the lemniscatic functions was an early step toward elliptic functions.

Infinite series

For the general reader. Add \(1+\tfrac12+\tfrac13+\tfrac14+\cdots\). The terms shrink to nothing, yet the total grows past any bound. Add instead \(1+\tfrac14+\tfrac19+\tfrac1{16}+\cdots\) and the total stays below 2. Bernoulli proved both facts in a series of five Basel dissertations from 1689 to 1704, and he asked for the exact value of the second sum. That question, the Basel problem, waited more than forty years for Euler.

The five dissertations, Positiones arithmeticae de seriebus infinitarum, were defended by students between 1689 and 1704; Jakob Hermann defended the third and Bernoulli’s nephew Nicolaus the fifth. They were reprinted together as an appendix to Ars Conjectandi.

Proof. (a) Group the terms in blocks: for each \(m\ge1\), \(\sum_{n=2^{m-1}+1}^{2^m}\frac1n\ge2^{m-1}\cdot\frac1{2^m}=\frac12\). There are infinitely many blocks, so the partial sums are unbounded. This grouping argument goes back to Nicole Oresme in the fourteenth century; Pietro Mengoli proved divergence in 1650. The Bernoulli brothers gave new proofs. (b) For \(n\ge1\), \(\frac1{n^2}\le\frac{2}{n(n+1)}=2\Big(\frac1n-\frac1{n+1}\Big)\), and the right side telescopes to a sum of 2. Strict inequality holds for \(n\ge2\).

Bernoulli could not find the sum of \(\sum 1/n^2\) and invited readers to send it to him. Euler found it, \(\pi^2/6\), in the 1730s, while working in St Petersburg alongside Jacob’s nephew Daniel.

Series with polynomial coefficients

The dissertations also sum series such as \(\sum n^kx^n\) by repeated manipulation of the geometric series. Differentiating \(\sum_{n\ge0}x^n=\frac1{1-x}\) and multiplying by \(x\) gives, for \(|x|<1\),

Sums of powers and the Bernoulli numbers

For the general reader. Everyone learns that \(1+2+\cdots+n=n(n+1)/2\). There are similar formulas for squares, cubes and higher powers, and mathematicians from Archimedes to Johann Faulhaber found many of them. In Ars Conjectandi Bernoulli gave one rule that produces them all. The rule depends on a single sequence of rational numbers, now called the Bernoulli numbers, which turned out to be among the most important numbers in mathematics.

Part II of Ars Conjectandi, on combinations, needs sums of the form \(\sum k^c\). Bernoulli listed the formulas for \(c=1,\dots,10\), noticed the pattern in their coefficients, and stated the general rule. The Japanese mathematician Seki Takakazu found the same numbers independently; his work was also published posthumously, in 1712.

This is Bernoulli’s own convention, \(B_1=+\tfrac12\). Many modern texts use \(B_1=-\tfrac12\) and the generating function \(\frac{t}{e^t-1}=\sum B_n\frac{t^n}{n!}\). With Bernoulli’s sign it reads \(\frac{t}{1-e^{-t}}=\sum B_n\frac{t^n}{n!}\).

Proof sketch. Let \(S_c(n)=\sum_{k=1}^{n}k^c\). Consider \(F(t)=\sum_{c\ge0}S_c(n)\frac{t^c}{c!}=\sum_{k=1}^{n}e^{kt}=\frac{e^{(n+1)t}-e^t}{e^t-1}=e^t\,\frac{e^{nt}-1}{e^t-1}\). Write \(\frac{t\,e^t}{e^t-1}=\frac{t}{1-e^{-t}}=\sum B_j\frac{t^j}{j!}\) and \(\frac{e^{nt}-1}{t}=\sum_{i\ge0}\frac{n^{i+1}t^i}{(i+1)!}\). Multiplying the two series and reading off the coefficient of \(t^c/c!\) gives the formula. The recursion for \(B_m\) is the case \(n=1\), where \(S_c(1)=1\).

Bernoulli was proud of the rule. He wrote that it had taken him less than half of a quarter of an hour to find the sum of the tenth powers of the first thousand numbers, and gave the 32-digit result (Exercise 8.3).

Later life of the numbers

Euler connected them to his solution of the Basel problem: \(\zeta(2k)=\sum_{n\ge1}n^{-2k}=\frac{(-1)^{k+1}B_{2k}(2\pi)^{2k}}{2(2k)!}\). They appear in the Euler–Maclaurin summation formula, in the Taylor series of \(\tan x\), and in Ernst Kummer’s work on Fermat’s last theorem, where the primes dividing a numerator of some \(B_{2k}\) are the irregular primes.

The art of conjecturing

For the general reader. Before Bernoulli, probability meant the arithmetic of fair games of dice and cards. He wanted to apply it to the uncertain questions of real life: illness, weather, law, the reliability of witnesses. His book Ars Conjectandi begins with games, builds the combinatorics needed to count outcomes, and ends with the claim that experience can measure probabilities that reason alone cannot compute.

The book

The book has four parts. Part I reprints Christiaan Huygens’s 1657 tract De ratiociniis in ludo aleae, the first printed treatise on probability, with Bernoulli’s commentary and solutions to Huygens’s five problems. Part II treats permutations and combinations, gives a proof of the binomial theorem for positive integer exponents, and contains the sums of powers of Lesson 08. Part III applies the theory to twenty-four problems on games of cards and dice. Part IV, on applications to civil, moral and economic matters, contains the golden theorem (Lesson 10) and is unfinished.

Expectation

Huygens’s basic principle is that a chance of obtaining \(a\) or \(b\) with equal ease is worth \((a+b)/2\). More generally, if outcomes worth \(a_1,\dots,a_m\) occur in \(n_1,\dots,n_m\) equally likely cases, the chance is worth \(\sum n_ia_i/\sum n_i\). This is the expectation of modern probability.

Repeated trials

Reason. Each particular sequence with \(k\) successes and \(n-k\) failures occurs in \(r^k(t-r)^{n-k}\) of the \(t^n\) equally likely cases, and there are \(\binom nk\) such sequences. Trials of this kind, with two outcomes and a fixed probability, are now called Bernoulli trials.

The golden theorem

For the general reader. An urn holds 3000 white and 2000 black pebbles, but you are not allowed to count them. Draw one, note its colour, put it back, and repeat. Bernoulli proved that with enough draws you can be practically certain that the proportion of white you observe is close to the true \(3/5\). He called this his golden theorem and said it had occupied him for twenty years. It is the first limit theorem of probability, now called the weak law of large numbers.

Bernoulli’s statement

Let the ratio of fertile to sterile cases be \(r:s\), with \(t=r+s\), so \(p=r/t\). Bernoulli proved that for any \(c>0\) one can find a number of trials \(N=nt\) such that it is more than \(c\) times as likely that the observed proportion of successes lies between \((r-1)/t\) and \((r+1)/t\) as outside those bounds. He called probabilities so close to 1 “moral certainty”. This is the problem of learning probabilities a posteriori, from results, when they cannot be found a priori, from a count of cases.

A modern proof

\(S_N\) has mean \(Np\) and variance \(Npq\). Chebyshev’s inequality (1867) gives

Bernoulli’s proof

Bernoulli had no variance. He worked directly with the terms of the expansion of \((r+s)^{nt}\), whose terms, divided by \(t^{nt}\), are the probabilities of each number of successes. The largest term sits at \(nr\) successes (Exercise 9.3). Using the ratio of consecutive terms, he compared the \(n\) terms just beyond \(nr+n\) with the \(n\) terms just inside, showed that the ratio between corresponding terms grows without bound with \(n\), and so bounded the total outside the window by a fraction \(1/c\) of the total inside. The argument is elementary and complete.

He then computed how many trials suffice. For \(r=30,\ s=20\) (so \(p=3/5\) and the window is \(29/50\) to \(31/50\)), he found that 25,550 trials make the window more than 1000 times as likely as its complement, 31,258 make it 10,000 times as likely, and 36,966 make it 100,000 times as likely. Each further factor of 10 costs 5,708 trials.

Bernoulli’s equation and the isoperimetric problem

For the general reader. Two of Jacob’s challenge problems became standard topics. One is a nonlinear differential equation that turns out to be linear in disguise. The other, his reply to Johann’s brachistochrone, asks for the best curve among those of a fixed length. Jacob’s solution of 1701 was correct where Johann’s first attempt was not, and its methods fed into the calculus of variations of Euler and Joseph-Louis Lagrange.

Bernoulli’s equation

In 1695 Jacob proposed in Acta Eruditorum the equation now named after him. Leibniz and Johann solved it, Johann by writing the unknown as a product of two functions; Jacob published his own solution in 1696. In modern form:

The isoperimetric problem

Jacob’s problem of 1697 asked, among curves of given length on a given base, for one that maximizes an area defined from the curve, in a class that includes the classical problem of Dido. The general shape of such problems, in modern language:

Jacob’s 1701 Analysis magni problematis isoperimetrici did not use a multiplier. He varied the curve at three consecutive points at once, so that the length could stay fixed, and found that in general the solution requires a third-order differential equation. He also proved there that the catenary is the curve of given length whose centre of gravity lies lowest (Exercise 11.2).

Brothers at war, 1691–1705

The partnership between the brothers lasted only a few years. Through the early 1690s they worked on the same problems, including caustics, the envelopes of reflected and refracted light rays, which Jacob treated for the parabola and the logarithmic spiral in 1692. But Johann resented being treated as a pupil, and Jacob resented his brother’s boasting. Johann left for Paris in 1691 and for a professorship at Groningen in 1695.

From 1696 the quarrel was public. When Johann posed the brachistochrone, Jacob solved it in 1697 by a method that, unlike Johann’s optical trick, generalized to other variational problems. He answered with the isoperimetric problem. Johann’s first answer was wrong, Jacob said so in print, and insults flew in the journals for years. When the Paris Academy of Sciences admitted both brothers in 1699, it reportedly did so on condition that they stop quarrelling in public. Jacob was also elected to the Berlin society founded by Leibniz.

Jacob was a careful, cautious teacher. His students included Jakob Hermann, who defended his third series dissertation, later held chairs at Padua, Frankfurt an der Oder and St Petersburg, and wrote his obituary in Acta Eruditorum in 1706. His nephew Nicolaus Bernoulli, later known as Nicolaus I, defended the fifth dissertation in 1704 and became a leading probabilist.

In his last years Bernoulli corresponded with Leibniz about the golden theorem. Leibniz doubted that a finite number of observations could ever determine a probability, since nature might change. Bernoulli held that, under stable conditions, the theorem settled the matter. He died in Basel on 16 August 1705, aged 50, with Ars Conjectandi unfinished. Johann, on his way back from Groningen, learned of his death and succeeded to the chair.

The posthumous book and the legacy

Ars Conjectandi was printed in Basel in 1713, eight years after Jacob’s death, prepared by his son Nicolaus, with a preface by his nephew Nicolaus I. The volume also included the five series dissertations and a letter in French on scoring in real tennis (jeu de paume). By then Pierre Rémond de Montmort and Abraham de Moivre had begun their own work on chance. De Moivre’s Doctrine of Chances (1718) built on Bernoulli, and in 1733 de Moivre sharpened the golden theorem with the normal approximation to the binomial, the first case of the central limit theorem.

The tombstone

Bernoulli asked for a logarithmic spiral to be carved on his tombstone with the words Eadem mutata resurgo, “changed, I shall arise the same”, a reference both to the spiral’s self-similarity and to the resurrection. The stone in the cloister of Basel Minster shows a spiral, but the mason carved an Archimedean spiral, whose turns are equally spaced, instead of a logarithmic one.

Legacy

• Probability and statistics: the first limit theorem, the idea of estimating unknown probabilities from frequencies, and Bernoulli trials and the Bernoulli distribution.
• Analysis: the first published use of “integral”, the study of \(e\) through compound interest, the Bernoulli numbers, and early convergence results for series.
• Differential equations and variational calculus: Bernoulli’s equation, the isoperimetric problem and his three-point variations, which Euler and Lagrange developed into the general theory.
• Mechanics: the elastica, whose curvature–moment law Euler made the basis of the theory of elastic beams.
• Family: he taught Johann, and through him the next generation, including Daniel Bernoulli and Euler.

Connections

People linked to Jacob Bernoulli across this collection. A name without a link has no page here yet. See Jacob Bernoulli in the network of all 337 lives →

Students

• Johann Bernoulli 1687-1691
• Jakob Hermann
• Nicolaus I Bernoulli 1704

Influenced by

• Christiaan Huygens
• Gottfried Wilhelm Leibniz
• René Descartes

Influenced

• Abraham de Moivre
• Andrey Markov
• Leonhard Euler
• Pierre-Simon Laplace

Colleagues and collaborators

• Gottfried Wilhelm Leibniz 1687-1705
• Robert Boyle
• Robert Hooke
• Johannes Hudde 1681

Family

• Daniel Bernoulli
• Johann Bernoulli
• Nicolaus I Bernoulli

Also mentioned in this life

• Archimedes
• Thomas Bayes
• Pafnuty Chebyshev
• Pierre de Fermat
• Carl Friedrich Gauss
• Ernst Eduard Kummer
• Joseph-Louis Lagrange
• Colin Maclaurin
• Blaise Pascal
• Evangelista Torricelli

Fields

• probability theory 23 lives
• calculus 9 lives
• infinite series 3 lives
• differential equations 13 lives
• calculus of variations 9 lives

On CoursesHub

• ODEs / PDEs / Green Functions · Physical Mathematics
• Variational Calculus · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Jacob Bernoulli, MacTutor History of Mathematics.
2. Jacob Bernoulli, Wikipedia.
3. J. O. Fleckenstein, Bernoulli, Jakob (Jacques) I, Dictionary of Scientific Biography, encyclopedia.com.
4. J. J. O’Connor and E. F. Robertson, Johann Bernoulli, MacTutor.

Mathematics

1. Jacob Bernoulli, Ars Conjectandi, Basel, 1713. English translation by E. D. Sylla, The Art of Conjecturing, Johns Hopkins University Press, 2006.
2. Jacob Bernoulli, Opera, 2 vols., Geneva, 1744.
3. B. Sung (trans.), Jacob Bernoulli, The Art of Conjecture, Part IV, Harvard technical report, 1966.
4. Ars Conjectandi, Wikipedia.
5. Bernoulli number and Lemniscate of Bernoulli, Wikipedia.
6. S. M. Stigler, The History of Statistics: The Measurement of Uncertainty before 1900, Harvard University Press, 1986, chapter 2.
7. A. Hald, A History of Probability and Statistics and Their Applications before 1750, Wiley, 1990.
8. C. Truesdell, The rational mechanics of flexible or elastic bodies 1638–1788, in Euler, Opera omnia ser. II vol. 11, 1960.