Bernoulli Family

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Bernoulli Family

SUPPLEMENTARY BIBLIOGRAPHY

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 ?x is proportional to ?x 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, seeStatistics, article onThe history of statistical method; and the biography ofMoivre; for discussion of the subsequent development of their ideas, seeProbability.]