Daniel Bernoulli

How to use this course

Daniel Bernoulli trained as a physician and spent his career applying mathematics to the physical and living world. His Hydrodynamica of 1738 gave fluid mechanics its name and its most famous principle, and contains the first kinetic theory of gases. He found the normal modes of a hanging chain, argued that every vibration of a string is a sum of simple ones, measured the value of a risky bet by its utility, and built the first mathematical model of an epidemic. He won the prize of the Paris Academy ten times. This course follows his physics through these topics, and his life through a family more famous for quarrels than for affection.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern notation, derivations, and exercises with folded solutions. Bernoulli’s own arguments were often physical and approximate; the lessons say where later work by Leonhard Euler, Jean le Rond d’Alembert and others made them exact. The companion courses on Johann and Jacob Bernoulli cover the family’s earlier mathematics.

Timeline

A son of Johann, 1700–1721

Daniel Bernoulli was born on 8 February 1700 in Groningen, in the Dutch Republic, where his father Johann held the chair of mathematics. His mother was Dorothea Falkner, from a Basel family. His elder brother Nicolaus, known as Nicolaus II, was born in 1695; a younger brother, Johann II, would also become a mathematician. In 1705 the family returned to Basel, where Johann took over the chair left vacant by the death of his brother Jacob.

Johann had been forced by his own father into medicine and had become a mathematician anyway. He did the same to his son. He decided that mathematics paid badly and tried to place Daniel in business. When that failed he allowed him to study medicine, while teaching him mathematics at home. Daniel entered the University of Basel at 13, took a bachelor’s degree in 1715 and a master’s in 1716. He studied medicine at Heidelberg in 1718 and at Strasbourg in 1719, and returned to Basel to take his doctorate in 1721. His dissertation, De respiratione, treated breathing as a mechanical process.

In 1721 he applied for the chair of anatomy and botany at Basel. Chairs there were filled partly by lot among the qualified candidates, and he lost the draw. He failed again for the chair of logic. These failures sent him abroad.

Venice and the Exercitationes, 1723–1725

In 1723 Bernoulli went to Venice to study practical medicine with the physician Pietro Antonio Michelotti. He had hoped to continue to Padua and work with the anatomist Giovanni Battista Morgagni, but illness kept him in Venice. There, with the help of Christian Goldbach, he published his first mathematical work, Exercitationes quaedam mathematicae (1724).

The little book covered four topics: the card game faro, the flow of water from a vessel, the differential equation of Jacopo Riccati, and figures bounded by circular arcs. On the Riccati equation \(y'+ay^2=bx^m\) he showed that the equation can be solved by elementary integration when \(m=-2\) or \(m=-4k/(2k\pm1)\) for a positive integer \(k\). A century later Joseph Liouville proved that these are the only such cases. In 1725 he also won his first prize from the Paris Academy of Sciences, for the design of an hourglass or water clock that would keep time on a ship at sea.

The Exercitationes brought an invitation from the Academy of Sciences that Peter the Great had founded at St Petersburg. His brother Nicolaus was invited too. The brothers set out together in 1725.

St Petersburg, 1725–1733

The two brothers arrived in St Petersburg in October 1725. Within a year Nicolaus died of a fever. His father’s former pupil, the young Leonhard Euler, was looking for a post; with the Bernoullis’ support the Academy invited him to fill a place in physiology, and Euler arrived in 1727. The two became lifelong friends and correspondents. Their letters, continued after Bernoulli left Russia, are a running record of eighteenth-century mechanics.

The St Petersburg years were the most creative of Bernoulli’s life. He worked on the mechanics of muscle contraction, on the oscillations of hanging chains and connected pendulums (Lesson 06), and on the theory of probability (Lesson 08). He wrote most of the Hydrodynamica there (Lesson 04). In 1726 he pointed out that the motion of a rigid body can be split into a translation of its centre of mass and a rotation about it, and in 1729 he published an iterative method for finding the largest root of a polynomial, still called Bernoulli’s method.

The Academy paid well, but Bernoulli disliked Russia and wanted a chair at home. He applied for Basel posts several times without success, and in 1732 he finally obtained the chair of anatomy and botany. He left St Petersburg in 1733 with his younger brother Johann II. On the way home they visited Danzig, Hamburg, the Netherlands and Paris.

Hydrodynamica: pressure and speed

For the general reader. Water squeezed through a narrow part of a pipe speeds up, and as it speeds up, its pressure drops. Bernoulli discovered this by experiment and explained it by the conservation of what was then called living force, an early form of energy. He called the subject hydrodynamica, a word he coined, and the result, now Bernoulli’s principle, explains everything from a garden sprayer to the lift on a wing.

The book

Hydrodynamica, sive de viribus et motibus fluidorum commentarii was largely written in St Petersburg, largely finished by 1734 and printed in Strasbourg in 1738. Its thirteen sections treat the outflow of water from vessels, the oscillation of water in tubes, the pressure of moving fluids, machines for raising water, the jet propulsion of ships and, in Section 10, the properties of elastic fluids, that is, gases (Lesson 05). Bernoulli’s basic tool was the principle that the actual descent of the centre of gravity of a system equals its potential ascent, the height to which the parts could rise if their velocities were turned upward. In modern words, kinetic plus potential energy is conserved.

Efflux

For water leaving a wide vessel through a small hole at depth \(h\) below the free surface, the energy balance gives the outflow speed \(v=\sqrt{2gh}\), the result found by experiment by Evangelista Torricelli. Bernoulli treated the corrections due to the finite width of the vessel and to the gradual start of the flow.

Pressure in moving water

Bernoulli’s great innovation was to ask for the pressure on the wall of a pipe while water flows through it. He measured it by piercing the wall and fitting a vertical glass tube: the height of the water in the tube gives the pressure. He found it lower where the flow was faster. The equation below is the form Euler gave in the 1750s from his general equations of fluid motion.

Derivation by energy. Follow a thin tube of streamlines between cross-sections 1 and 2. In a short time a volume \(\delta V\) enters at 1 and the same volume leaves at 2, by incompressibility. The pressure forces do work \((p_1-p_2)\,\delta V\) on the fluid between the sections. The flow is steady, so the only change in energy is that a mass \(\rho\,\delta V\) has been moved from section 1 to section 2, changing its kinetic and potential energy by \(\rho\,\delta V\big(\tfrac12v_2^2-\tfrac12v_1^2+g z_2-gz_1\big)\). With no friction, work equals change of energy, which gives the theorem.

The first kinetic theory of gases

For the general reader. Why does a gas push on its container? In Section 10 of Hydrodynamica Bernoulli imagined air as a swarm of tiny particles in rapid motion, bouncing off the walls. From this picture he derived Boyle’s law, that pressure is inversely proportional to volume, and predicted that pressure grows with the square of the particles’ speed, so with heat. The idea was largely ignored for a century, until Rudolf Clausius, James Clerk Maxwell and Ludwig Boltzmann built statistical mechanics on it.

Bernoulli’s model

Bernoulli considered a vertical cylinder closed by a weighted piston, containing very small corpuscles moving rapidly in all directions. The weight on the piston is held up by the impacts of the particles. He argued that if the volume is reduced, the particles are closer together and strike the piston more often, so that the pressure rises in inverse proportion to the volume, with a correction for the particles’ own size. He also argued that the pressure is proportional to the square of the particles’ speed, because a faster particle hits both harder and more often.

Derivation. Take a cubical box of side \(L\). A particle with velocity component \(v_x\) hits the wall \(x=L\) every \(2L/|v_x|\) seconds and each time transfers momentum \(2m|v_x|\). Its average force on the wall is \(2m|v_x|\cdot|v_x|/(2L)=mv_x^2/L\). Summing over particles, the pressure is \(\sum mv_x^2/L^3=Nm\overline{v_x^2}/V\). Isotropy gives \(\overline{v_x^2}=\tfrac13\overline{v^2}\).

At fixed \(\overline{v^2}\), \(pV\) is constant: Boyle’s law. Combined with the ideal gas law \(pV=Nk_BT\), the theorem says that the mean kinetic energy of a molecule is \(\tfrac32k_BT\), which identifies temperature with molecular motion, as Bernoulli suggested.

The hanging chain and its modes

For the general reader. Hang a chain from a hook and swing it gently. It can swing in a single bow, or in an S-shape, or in shapes with more wiggles, each with its own frequency. Around 1732 Bernoulli found these special shapes, the first normal modes ever computed for a continuous body. To describe them he wrote down a power series that is now called a Bessel function.

Bernoulli first studied a weightless thread carrying two, three and more equal weights, and found special motions in which all the weights swing in step with a common frequency, as many such motions as there are weights. Letting the number of weights grow, he passed to the uniform chain. The work appeared in the St Petersburg Commentarii under the title Theoremata de oscillationibus corporum filo flexili connexorum et catenae verticaliter suspensae.

The equation

Let a uniform chain of length \(L\) hang from a fixed point, and measure \(x\) upward from the free end. The tension at height \(x\) supports the chain below it, so \(T=\mu gx\), where \(\mu\) is the mass per length. For small horizontal displacements \(y(x,t)\), Newton’s law for a short piece gives

Derivation. Substitute \(y=f(x)\cos\omega t\): \(g\,(xf')'+\omega^2f=0\). Seek \(f=\sum a_kx^k\). Then \((xf')'=\sum k^2a_kx^{k-1}\), and matching powers gives \(g(k+1)^2a_{k+1}=-\omega^2a_k\), so \(a_k=(-1)^k\frac{(\omega^2/g)^k}{(k!)^2}a_0\). With \(z=2\omega\sqrt{x/g}\) we have \((\omega^2x/g)^k=(z/2)^{2k}\), which is the series for \(J_0(z)\). This is Bernoulli’s series. The condition at the hook, \(J_0(2\omega\sqrt{L/g})=0\), gives the frequencies. The second solution of the equation is unbounded at \(x=0\) and is excluded. Euler took up the problem again in 1781, and Friedrich Bessel studied these functions systematically in 1824.

Vibrating strings and superposition

For the general reader. A plucked string sounds a note and, more faintly, its overtones. Bernoulli claimed that every motion of the string is simply all of these pure vibrations sounding together. Euler and d’Alembert disagreed, and the argument, about what kind of functions a series of sines can represent, lasted until Joseph Fourier’s work in the nineteenth century and beyond.

The wave equation and its modes

In 1747 d’Alembert derived the equation of a vibrating string, \(y_{tt}=c^2y_{xx}\) with \(c^2=T/\mu\), and gave its general solution \(y=F(x+ct)+G(x-ct)\). Euler extended it in 1748 to arbitrary initial shapes, including shapes with corners. With fixed ends \(y(0,t)=y(L,t)=0\), separation of variables gives the modes

Bernoulli argued physically: the sound of a string is a mixture of its harmonics, and with infinitely many free coefficients \(a_n\) one can fit any shape. Euler objected that a sum of sines is periodic and smooth, while a plucked string starts with a corner, so the series could not represent all initial shapes. D’Alembert went further and doubted even Euler’s generality. The question, which functions equal the sum of their trigonometric series, was answered for wide classes of functions by Fourier (1807–1822) and Peter Gustav Lejeune Dirichlet (1829), and in its general form it led to Bernhard Riemann’s integral and Georg Cantor’s set theory. In modern terms Bernoulli was right: every square-integrable initial shape, corners included, has a convergent sine expansion.

Measuring risk: the St Petersburg game

For the general reader. A coin is tossed until it lands heads. If that happens on the first toss you win 1 ducat, on the second 2, on the third 4, doubling each time. The average prize is infinite, yet no sensible person would pay more than a few ducats to play. Bernoulli resolved the puzzle in 1738: what matters is not money but its usefulness to you, and an extra ducat matters less to the rich than to the poor. This idea, expected utility, became the foundation of the economics of risk.

The problem was posed by Jacob’s nephew Nicolaus I Bernoulli in a letter to Pierre Rémond de Montmort in 1713. Daniel’s solution, Specimen theoriae novae de mensura sortis, appeared in the St Petersburg Commentarii for 1730–31, printed in 1738; the game has carried the city’s name ever since. In the paper Daniel acknowledged that Gabriel Cramer had reached a similar solution independently in 1728.

The paradox

The prize is \(2^{k-1}\) ducats with probability \(2^{-k}\), so the expected prize is \(\sum_{k\ge1}2^{-k}\,2^{k-1}=\tfrac12+\tfrac12+\cdots=\infty\).

Under this hypothesis a person with wealth \(w\) values the free gift of the game at the certainty equivalent \(c(w)\) defined by

The series converges, since its terms are of order \(k\,2^{-k}\). For \(w=0\) the right side is \(\sum 2^{-k}(k-1)\ln2=\ln2\), so the game is worth 2 ducats. For \(w=1000\), \(c\approx6\) ducats, the value Bernoulli gave. Even a millionaire should value it at only about 11 ducats.

Smallpox and the first epidemic model

For the general reader. In the eighteenth century smallpox killed a large fraction of each generation. Inoculation, deliberately infecting a healthy person with matter from a mild case, gave immunity but carried a small risk of death. Was it worth it? In 1760 Bernoulli answered with a mathematical model, the first in epidemiology, and calculated that inoculating everyone at birth would add about three years to average life expectancy.

The model

Bernoulli followed a single cohort of newborns. Let \(w(a)\) be the number still alive at age \(a\), and \(s(a)\) the number of them who have not yet had smallpox. He assumed that each susceptible person catches smallpox at rate \(q\) per year, that a fraction \(p\) of those infected die of it, and that the rest become immune for life. Let \(\mu(a)\) be the death rate from all other causes, the same for everyone. Then

Derivation. \(\xi'=\frac{s'}{w}-\frac{sw'}{w^2}=-q\xi-\mu\xi+pq\xi^2+\mu\xi=-q\xi+pq\xi^2\). The term \(\mu\) cancels. This is a Bernoulli equation (named after Daniel’s uncle Jacob) with exponent 2, so set \(z=1/\xi\): \(z'=qz-pq\), with solution \(z=p+(1-p)e^{qa}\).

Bernoulli took \(p=q=\tfrac18\), from the observation that about one person in eight infected died and that the young caught the disease at about that rate. He combined \(\xi(a)\) with Edmond Halley’s 1693 life table for Breslau to compute \(\mu\) and then the survival curve in a world without smallpox. Life expectancy at birth rose from about 26 years 7 months to 29 years 9 months. He read the memoir to the Paris Academy in 1760, and it was published in 1766. D’Alembert criticized the assumptions and argued that people value a present risk of death differently from a later one, a point about decision-making that Bernoulli’s own work on utility had raised.

Errors, likelihood and chance in the sky

For the general reader. Astronomers repeat a measurement several times and get slightly different answers. Which value should they report? The usual answer was the average. In 1778 Bernoulli proposed choosing the value that makes the observed measurements most probable, the principle now called maximum likelihood. Forty years earlier he had used probability to ask whether the planets’ nearly flat arrangement could be an accident.

Maximum likelihood (1778)

Bernoulli’s memoir Dijudicatio maxime probabilis plurium observationum discrepantium (written 1777, published 1778) argued that small errors are more likely than large ones and that errors beyond some limit \(r\) do not occur. He modelled the error density by a semicircle and proposed that the best estimate is the value that maximizes the product of the densities of the observed errors. Euler, in a commentary, agreed with the principle but proposed a different calculation.

For a normal error law, \(\ln L=-\sum(x_i-\theta)^2/(2\sigma^2)+\text{const}\), and the maximum is the mean. For other laws it is not. Bernoulli’s semicircle gives an estimate that weights observations near the centre more heavily than outlying ones. Carl Friedrich Gauss later reversed the logic: he asked for the error law under which the mean is the maximum likelihood estimate and found the normal distribution (1809). Ronald Fisher made maximum likelihood the central method of statistics in the 1920s.

The orbits of the planets (1734)

The Paris Academy’s prize question for 1734 asked for the physical cause of the small inclinations of the planetary orbits to one another. In his prize essay, Bernoulli asked first whether the closeness of the orbital planes could be due to chance. Taking the planes as independent and random, he computed the probability that all of them would fall within the observed small angle of a reference plane, found it extremely small, and concluded that a common cause was needed. The argument is an early example of what is now called a test of significance.

Basel, prizes and quarrels, 1733–1782

Bernoulli took up the chair of anatomy and botany in Basel at the end of 1733. In 1734 the Paris Academy divided its Grand Prize, on the inclinations of planetary orbits, between him and his father. Johann, who had expected to win alone, was so angry at being rated his son’s equal that he barred Daniel from his house. 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, and Daniel complained bitterly that he had been robbed of his whole work. The two were never reconciled.

The prizes kept coming. Bernoulli won or shared the Paris prize ten times between 1725 and 1757: on the hourglass for ships (1725), the planetary orbits (1734, with his father), the shape of anchors (1737, with Giovanni Poleni), the tides (1740, shared with Euler, Colin Maclaurin and Antoine Cavalleri), magnetism (1743 and 1746), determining time at sea (1747), ocean currents (1751), the action of forces on ships (1753), and the reduction of a ship’s pitching and rolling (1757). The tides memoir developed Isaac Newton’s equilibrium theory in detail.

His teaching followed his interests. In 1743 he exchanged botany for physiology, and in 1750 he took the chair of physics, which he held until 1776. He was elected a Fellow of the Royal Society in 1750 and belonged to the leading academies of Europe. He continued to correspond with Euler, who used Bernoulli’s suggestion that the elastic energy of a bent beam is proportional to \(\int ds/R^2\) as the starting point for his 1744 theory of elastic curves; the Euler–Bernoulli beam equation honours both.

His late work turned to probability and statistics: the smallpox memoir, studies of the durations of marriages, and the 1778 memoir on observations. He died in Basel on 17 March 1782, aged 82, and was buried in the Peterskirche.

Legacy

Bernoulli was a physicist in the modern sense before the word existed: he began from experiment or observation, built a simplified mathematical model, solved it, and compared it with the data. His methods were often less rigorous than Euler’s, and Euler, d’Alembert and Joseph-Louis Lagrange turned several of his insights into general theories. But the insights were his.

• Fluid mechanics: the word hydrodynamics, the pressure–speed relation that bears his name, and the energy method for flows. Euler’s equations of 1757 grew out of Hydrodynamica.
• Kinetic theory: the first explanation of gas pressure by molecular motion, revived by John Herapath, John James Waterston, Clausius and Maxwell in the nineteenth century.
• Vibrations: the first normal modes of a continuous system, the first Bessel function, and the principle of superposition, the seed of Fourier analysis.
• Probability and economics: expected utility and risk aversion, revived by John von Neumann and Oskar Morgenstern in 1944; maximum likelihood; and an early significance test.
• Epidemiology: the first compartmental model of an infectious disease, a direct ancestor of the models used in public health today.

Connections

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

Teachers and advisors

• Johann Bernoulli

Influenced by

• Brook Taylor
• Edmond Halley
• Isaac Newton
• Robert Boyle

Influenced

• Carl Friedrich Gauss
• Joseph Fourier
• Joseph-Louis Lagrange
• Ludwig Boltzmann

Colleagues and collaborators

• Colin Maclaurin
• Leonhard Euler
• Christian Goldbach
• Gabriel Cramer

Rivals

• Jean le Rond d'Alembert

Friends

• Leonhard Euler 1727-1782
• Pierre-Louis Moreau de Maupertuis

Family

• Jacob Bernoulli
• Johann Bernoulli
• Johann II Bernoulli
• Nicolaus I Bernoulli
• Nicolaus II Bernoulli

Also mentioned in this life

• Friedrich Wilhelm Bessel
• Georg Cantor
• Rudolf Clausius
• Peter Gustav Lejeune Dirichlet
• Ronald Fisher
• Pierre-Simon Laplace
• Joseph Liouville
• James Clerk Maxwell
• Bernhard Riemann
• Evangelista Torricelli
• Johannes Diderik van der Waals
• John von Neumann

Fields

• fluid mechanics 6 lives
• kinetic theory 4 lives
• probability theory 23 lives
• medicine 8 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Daniel Bernoulli, MacTutor History of Mathematics.
2. Daniel Bernoulli, Wikipedia.
3. H. Straub, Bernoulli, Daniel, Dictionary of Scientific Biography, encyclopedia.com.
4. Bernoulli, Daniel, Neue Deutsche Biographie, deutsche-biographie.de.
5. Bernoulli, Daniel, Encyclopedia of Mathematics.

Physics and probability

1. D. Bernoulli, Hydrodynamica, Strasbourg, 1738. English translation by T. Carmody and H. Kobus, Hydrodynamics by Daniel Bernoulli and Hydraulics by Johann Bernoulli, Dover, 1968.
2. D. Bernoulli, Specimen theoriae novae de mensura sortis, Commentarii Academiae Scientiarum Imperialis Petropolitanae 5 (1738). English translation by L. Sommer, Exposition of a new theory on the measurement of risk, Econometrica 22 (1954).
3. K. Dietz and J. A. P. Heesterbeek, Daniel Bernoulli’s epidemiological model revisited, Mathematical Biosciences 180 (2002).
4. Bernoulli’s hanging chain problem: historical note, ProofWiki; G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge, 1922, §1.1.
5. C. Truesdell, The rational mechanics of flexible or elastic bodies 1638–1788, in Euler, Opera omnia ser. II vol. 11, 1960.
6. S. M. Stigler, The History of Statistics, Harvard University Press, 1986.
7. D. Bernoulli, The most probable choice between several discrepant observations (1778), English translation with Euler’s commentary in Biometrika 48 (1961).