## Bernoulli

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## Bernoulli Family

# Bernoulli Family

The Bernoullis, a Swiss family, acquired its fame in the history of science by producing eight or nine mathematicians of the first rank within three generations. They were all descendants of Niklaus Bernoulli, a prominent merchant in the city of Basel. Each of these mathematicians was compelled by his parents to study for one of the established professions before being permitted to embark upon his real interest, mathematics. Within the group there were four in particular who contributed to the theory of probability and mathematical statistics: Jakob (Jacques) I, Johann (Jean) I, Niklaus (Nicolas) I, and Daniel I.

## Jakob i and Johann i

The first in the line of the Bernoulli mathematicians, Jakob I (1654–1705), was the son of the merchant Niklaus. He completed theological studies and then spent six years traveling in England, France, and Holland. Returning to Basel, he lectured on physics at the university until he was appointed professor of mathematics in 1687. His younger brother, Johann I (1667–1748), studied for a medical degree, at the same time receiving instruction in mathematics from Jakob. Later Johann, too, became a professor of mathematics, teaching at the University of Groningen in Holland until he returned to Basel as Jakob’s successor.

The brothers were inspired by the works of Leibniz on the infinitesimal calculus, and they became his chief protagonists on the Continent. The new methods enabled them to solve an abundance of mathematical problems, many with applications to mechanics and physics. They applied differentiation and integration to find the properties of many important curves: they determined the form of the catenary curve (hanging chain) and the isochrone, or tautochrone (cycloid), and the form of a sail subject to wind pressure. Jakob was particularly fascinated by the logarithmic spiral, which he requested be engraved on his tombstone. Both used infinite series as a tool; the Bernoulli numbers were introduced by Jakob.

Johann was perhaps even more productive as a scientist than was Jakob. He studied the theory of differential equations, discovered fundamental principles of mechanics, and wrote on the laws of optics. Although the first textbook on the calculus, *Analyse des infiniment petits* (1696), was written by Antoine de L’Hospital, it was largely based upon the author’s correspondence with Johann.

The personal relations between the brothers were marred by violent public strife, mainly disputes about priority in the discovery of scientific results. Particularly bitter was their controversy over the brachystochrone, the curve of most rapid descent of a particle sliding from one point to another under the influence of gravity. The problem was of great theoretical interest, since it raised for the first time a question whose solution required the use of the principles of the calculus of variation (the solution is a cycloid).

During his stay in Holland, Jakob became interested in the theory of probability. In his lifetime he published very little on the subject—only a few scattered notes in the *Acta eruditorum.* His main work on probability, the *Ars conjectandi* (“The Art of Conjecturing”), was printed posthumously in Basel in 1713. It is divided into four books. The first is an extensive commentary upon Huygens’ pioneer treatise: “De ratiociniis in ludo aleae” (1657; “On Calculations in Games of Chance”). The second gives a systematic presentation of the theory of permutations and combinations, and in the third this is applied to a series of contemporary games, some quite involved. For each, Jakob computed the mathematical expectations of the participants.

The fourth book shows the greatest depth. Here Jakob tried to analyze the events to which probability theory is applicable; in other words, he dealt with the basic question of mathematical statistics: when is it possible to determine an unknown probability from experience? He emphasized that a great number of observations are necessary. Furthermore, he pointed out “something which perhaps no one has thought of before,” namely, that(in modern terminology) it is necessary to prove mathematically that as the number of observations increases, the relative number of successes must be within an arbitrarily small (but fixed) interval around the theoretical probability with a probability that tends to 1. This he did, with complete rigor and without the use of calculus, by examining the binomial probabilities and estimating their sums. Illustrations with numerical computations for small intervals are given. The author concluded with some philosophical observations which show the importance he attached to his theorem.

To the *Ars conjectandi* Jakob added a supplement on the *jeu de paume* (similar to the game of tennis), in the form of a letter to a friend. Here he computed the chances of winning for a player at any stage of the game, given players with equal skill and players with differing skill, and in the latter cases he determined how great an advantage the more skilled one can allow the other.

## Niklaus I

Niklaus I (1687–1759) was a nephew of Jakob I and Johann I; his father was a portrait painter. True to the family tradition, Niklaus studied for one of the older professions, jurisprudence, while on the side he attended the lectures in mathematics of his two uncles. His law thesis straddled both fields: “… de usu artis conjectandi in jure” (1709). He accepted a professorship in mathematics at Padua in 1716 but disliked the university there and returned to Basel in 1719. In 1722 he was appointed professor of logic; in 1731 he changed to a chair of jurisprudence.

When Jakob I died, his *Ars conjectandi* was not in finished form and the publisher asked Johann I to serve as editor. When Johann refused, Niklaus was suggested. He refused also, doubting his competence, but he was finally prevailed upon to accept the undertaking. Niklaus published little in the field of mathematics, probably because of his excessive modesty. But, as editor of the *Ars conjectandi*, he entered into extensive correspondence with the two other pioneers in probability, Rémond de Montmort and Abraham de Moivre. Both appealed for his support in the priority feud that arose between them.

Niklaus also corresponded with the Dutch physicist van s’Gravesande on a curious statistical phenomenon that had first been pointed out by Arbuthnot. It was generally accepted that births of boys as compared to girls correspond to a game of chance, with the same probability, *p* = 1/2, for each. Nevertheless, the birth records in London showed that for 82 successive years there had been more males born than females, a most unlikely occurrence under the assumption of equal probabilities. Van s’Gravesande and Arbuthnot were inclined to see this as an example of divine intervention in the laws of nature, while Niklaus took the view that it was more rational to assume that the probability for the birth of a male child is slightly greater than one-half.

## Daniel i

Johann I had three sons who were mathematicians: Niklaus II (1695–1726), Daniel I (1700–1782), and Johann II (1710–1790). He compelled each one to acquire a professional degree. Niklaus II studied law and began his career in Berne, as a professor in this subject, in 1723. In 1725 he was appointed to a professorship of mathematics at the Imperial Academy in St. Petersburg, but he died shortly after his arrival there. Johann II also studied jurisprudence; eventually he became his father’s successor as professor of mathematics in Basel. He continued the Bernoulli dynasty, having three sons who were mathematicians: Johann III, Daniel II, and Jakob II.

Daniel I studied medicine, but his first mathematical book had already appeared when he was 24 years old and the next year he was called to a mathematical professorship at the Imperial Academy in St. Petersburg, remaining there from 1725 to 1733. Upon his return to Basel he became professor of medicine and botany; in 1750 he was appointed to a professorship in physics, which suited him better.

Daniel I was a prolific writer, even by the standards of the Bernoulli family; no less than ten times were his works awarded prizes by the French Academy of Sciences. His main interests centered in theoretical physics, the foundation of mechanics and, later, probability. Some of his best-known papers deal with celestial mechanics, the tides, and the laws governing a vibrating string.

Most important among the papers on probability by Daniel I is the study *Specimen theoriae novae de mensura sortis* (1738). The basis for this work is the well-known Petersburg paradox, which at that time was a much discussed topic in connection with the concept of expected value. The paradox was first mentioned by Niklaus I in his correspondence with Montmort and is reproduced in the 1713 edition of Montmort’s book *Essay d’analyse sur les jeux de hazard.* The fact that the expectation is infinite led Daniel to introduce a moral expectation or marginal utility, now fundamental in economic investigations. He assumed that for a person with a fortune of size *x*, the utility of an increase Δx is proportional to Δx and inversely proportional to *x*, giving an expression

*u = a* log *x + b*

for the utility. In the same paper Daniel also pointed out that a similar idea had already been proposed by the Swiss mathematician G. Cramer in a letter of 1728 to Niklaus i. Daniel also wrote a few other papers on probability, but they are of lesser importance; a number of them are concerned with questions arising from mortality statistics.

Oystein Ore

[For *the historical context of the Bernoullis’ work, see*Statistics, *article on*The history of statistical method; *and the biography of*Moivre; *for discussion of the subsequent development of their ideas, see*Probability.]

## WORKS BY DANIEL I

1724 *Exercitationes quaedam mathematicae.* Venice (Italy): Apud Dominicum Lovisam.

(1738) 1954 Exposition of a New Theory on the Measurement of Risk. *Econometrica* 22:23–36. → First published as “Specimen theoriae novae de mensura sortis.”

## WORKS BY JAKOB I

(1713) 1899 *Wahrscheinlichkeitsrechnung (Ars conjectandi).* 2 vols. Leipzig: Engelmann. → First published posthumously in Latin.

1744 *Jacobi Bernoulli* … Opera. 2 vols. Geneva: Cramer & Fratrum Philibert. → Published posthumously.

## WORKS BY JOHANN I

1742 *Johannis Bernoulli … Opera omnia.* 4 vols. Geneva: Bousquet.

*Der Briefwechsel von Johann Bernoulli.* Volume 1. Basel: Birkhauser, 1955.

## WORKS BY NIKLAUS I

1709 *Dissertatio inauguralis mathematico-juridica de usu artis conjectandi in jure.* Basel: Mechel.

## SUPPLEMENTARY BIBLIOGRAPHY

Huygens, Christiaan 1657 De ratiociniis in ludo aleae. Pages 521–534 in Frans van Schooten, *Exercitationum mathematicarum libri quinque.* Leiden (Netherlands): Elsevier.

L’hospital, Guillaume François Antoine de 1696 *Analyse des infiniment petits, pour I’intelligence des lignes courbes.* Paris: Imprimerie Royale.

[Montmort, Pierre RÉmond de] (1708) 1713 *Essay d’analyse sur les jeux de hazard.* 2d ed. Paris: Quillau. → Published anonymously.

## Bernoulli Family

# Bernoulli Family

*Swiss Mathematicians*

*Seventeenth and Eighteenth Centuries*

The Bernoulli family was one of the world's most outstanding mathematical families. The members of the family who made the most significant contributions were two brothers, Jakob (1654–1705) and Johann (1667–1748), and Johann's son, Daniel (1700–1782).

## Family History

Jakob and Johann were among the ten children of a spice merchant from Basel, Switzerland. Jakob was forced by his father to study theology but refused a church appointment when he completed his doctorate. Instead, he accepted a mathematics position at the University of Basel in 1687, a position he held for the remainder of his life.

Johann, required to study medicine, entered the University of Basel in 1683, where his brother Jakob was already a professor. The writings of Gottfried Leibniz (1646–1716), introducing the new field of **calculus** , had proved to be too difficult for most mathematicians to understand. Jakob, however, mastered their obscurity and taught Johann. The Bernoullis subsequently were among the first to recognize the great potential of calculus and to present numerous applications of this new mathematics, extending its usefulness and popularity.

In 1691, Johann taught calculus to the prominent French mathematician L'Hôpital (1661–1704). Subsequently L'Hôpital published the first textbook on differential calculus, based entirely on Johann's notes but without giving him proper credit. Johann received his doctorate in medicine in 1694 and became professor of mathematics at the University of Gröningen in the Netherlands.

## Problems in the Family

Both brothers were making important original contributions and had risen to the top ranks of mathematicians, but they became jealous of each other. Jakob could not accept Johann, whom he had taught, as an equal, and Johann was unable to be professionally gracious. They attacked each other's work publicly. When Jakob died in 1705, Johann returned to Basel to become professor of mathematics, a position he held until his death in 1748.

Jakob Bernoulli was first to use the term "integral" in calculus. He and Johann introduced the calculus of variation. He solved the equation now known as **Bernoulli's Equation** and applied the methods of calculus to the problems of bridge design.

Jakob's highly significant work, *Ars Conjectandi,* was published in 1713, after his death. It included the first thorough treatment of probability, a discussion of Bernoulli's law of large numbers, the theory of permutations and combinations, a derivation of the exponential series using the Bernoulli numbers, and a discussion of mathematical and moral predictability.

Johann Bernoulli developed the theory of differential equations and discovered the Bernoulli series. His influential texts on integral calculus and differential calculus were published in the early 1740s. He applied the methods of calculus to numerous practical problems, including a number of aspects of navigation, optics, and kinetic energy. In addition to his influence on L'Hôpital, he taught the noted mathematician Leonhard Euler (1707–1783).

## Continuing the Legacy

Johann's son Daniel was born in 1700. Forced by his father to study medicine, he received a medical degree in 1721. After publishing a discussion of

differential equations and the properties of flowing liquids in 1724, he accepted a position at the Academy of Sciences in St. Petersburg, Russia, where he taught medicine and physics. He returned to Basel in 1732 to teach anatomy and botany.

In 1738, Daniel published *Hydrodynamica,* a thorough treatment of the properties of flowing fluids and the relationships among them. As a result, he is regarded as the founder of the science of **hydrodynamics** . He also treated the behavior of gases using mathematical probability, introducing the concepts that led to the **kinetic theory of gases** . He developed the mathematics and physics of vibrating strings and made numerous other contributions in a variety of fields. He accepted positions at Basel in physiology in 1743 and physics in 1750. He died in 1782.

A rather extraordinary incident occurred in 1735 when Johann and Daniel were awarded prizes by the Paris Academy of Sciences for separate work on planetary orbits. Johann became enraged at having to share the recognition with his son and banned him from his house. In 1738, when Daniel published *Hydrodynamica,* Johann published *Hydraulica,* which obviously plagiarized Daniel's work, apparently out of jealousy of his son's increasing reputation.

Other members of the Bernoulli family who were significant mathematicians included Johann's other two sons, Nikolaus (1695–1726), who died soon after accepting a position at the St. Petersburg Academy, and Johann II (1710–1790), who succeeded his father as professor of mathematics at Basel; Johann's nephew, Nicolaus (1687–1759), professor of mathematics at Padua; and Johann II's son, Johann III (1744–1807), who became a professor of mathematics at the Berlin Academy at the age of 19. Other grandsons and great grandsons made lesser contributions to mathematics.

see also Calculus; Euler, Leonhard.

*J. William Moncrief*

## Bibliography

Bell, E. T. "Nature or Nurture? The Bernoullis." In *Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré.* New York: Simon and Schuster, 1986.

Sensenbaugh, Roger. "The Bernoulli Family." In *Great Lives from History: Renaissance to 1900 Series.* Englewood Cliffs, NJ: Salem Press, 1989.

## Bernoulli

Bernoulli or Bernouilli (both: bĕrnōōyē´), name of a family distinguished in scientific and mathematical history. The family, after leaving Antwerp, finally settled in Basel, Switzerland, where it grew in fame. **Jacob, Jacques, or James Bernoulli,** 1654–1705, became professor at Basel in 1687. One of the chief developers both of the ordinary calculus and of the calculus of variations, he was the first to use the word *integral* in solving Leibniz's problem of the isochronous curve. He wrote an important treatise on the theory of probability (1713) and discovered the series of numbers that now bear his name, i.e., the coefficients of the exponential series expansion of *x*/(1-*e*^{-x}). He was succeeded at Basel by his brother, **Johann, Jean, or John Bernoulli,** 1667–1748, who earlier had been professor at Gröningen and who was famous for his work in the field of integral and exponential calculus and was also a founder of the calculus of variations. He also contributed to the study of geodesics, of complex numbers, and of trigonometry. His collected works were published under the title *Johannis Bernoulli opera omnia.* His son, **Daniel Bernoulli,** 1700–1782, was a mathematician, physicist, and physician and has often been called the first mathematical physicist. He received his doctorate in medicine but became professor of mathematics at the St. Petersburg Academy in 1725. He was professor of anatomy and botany at Basel from 1733, later becoming professor of natural philosophy (physics). His greatest work was his *Hydrodynamica* (1738), which included the principle now known as Bernoulli's principle, and anticipated the law of conservation of energy and the kinetic-molecular theory of gases developed more than 100 years later. He also made important contributions to probability theory, astronomy, and the theory of differential equations (solving a famous equation proposed by Riccati). Among the other noted members of the family are Nicolaus Bernoulli, 1662–1716, brother of Jacob and Johann, who was professor of mathematics at St. Petersburg; Nicolaus Bernoulli, 1695–1726, son of Johann and brother of Daniel, also a mathematician; Johann Bernoulli, 1710–90, another son of Johann (1667–1748) and brother of Daniel, who succeeded his father in the chair of mathematics at Basel and also contributed to physics; his son, Johann Bernoulli, 1746–1807, who was astronomer royal at Berlin and also studied mathematics and geography; and Jacob Bernoulli, 1759–89, another son of Johann (1710–90), who succeeded his uncle Daniel in mathematics and physics at St. Petersburg but met an early death by drowning.