The Bernoulli Family

Swiss mathematicians

• Daniel Bernoulli
• Born: February 8, 1700
• Birthplace: Groningen, United Provinces (now in the Netherlands)
• Died: March 17, 1782
• Place of death: Basel, Swiss Confederation (now in Switzerland)

• Jakob I Bernoulli
• Born: January 6, 1655 (December 27, 1654, old style)
• Birthplace: Basel, Swiss Confederation (now in Switzerland)
• Died: August 16, 1705
• Place of death: Basel, Swiss Confederation (now in Switzerland)

The Bernoulli family contributed to the flowering of mathematical analysis in the eighteenth century that applied advanced mathematical techniques to problems arising in physics, technology, medicine, and the emerging field of probability theory. Members of the family dominated continental mathematics from the later seventeenth to the later eighteenth centuries.

Early Lives

Jakob I, Johann I, and Daniel Bernoulli are the most important members of the Bernoulli Dynasty, but at least five other family members went on to achieve recognition from their contemporaries for their mathematical talents. There were so many Jakobs and Johanns that it has become standard to place Roman numerals after their names to help keep their identities clear. Johann I and Jakob I were brothers, and Daniel was the son of Johann I.

The Bernoullis were descended from a line of merchants. Johann I and Jakob I’s grandfather moved to Basel in 1622 and continued his profession as a druggist. His son, Nikolaus, became a minor local official. Jakob received his theological degree in 1676, while studying mathematics against his father’s wishes. He traveled extensively: He spent two years in Geneva as a tutor, then went to France to learn René Descartes’s approach to natural philosophy. He traveled to England in 1681, meeting Robert Hooke and Robert Boyle . He settled down somewhat in 1683, giving lectures, writing papers, and teaching himself more mathematics. He became a professor of mathematics at the University of Basel in 1687 and made himself master of the newly developed Leibnizian methods of infinitesimal mathematics.

Johann failed as an apprentice and received his father’s permission to enter the University of Basel in 1683, where Jakob had just begun lecturing. He began to study medicine in 1685, receiving his doctorate in 1694 for a mathematical account of the motion of muscles. Before receiving an offer for a post in 1695, Johann studied mathematics with Jakob and both became quite expert at Leibnizian calculus. Johann left for the chair of mathematics at Groningen, the Netherlands. In 1700, Daniel Bernoulli was born to Johann. Daniel obtained his master’s degree in 1716 and was taught mathematics by his father and his elder brother Nikolaus II. Attempts to place him as a commercial apprentice failed, and he studied medicine at several different universities, at last settling in Basel with a doctorate in 1721, his thesis concerning respiration. His first attempts to obtain a university post failed, but his Exercitationes quaedam mathematicae (1724; mathematical exercises) landed him a post at the St. Petersburg Academy.

Lives’ Work

Jakob became professor of mathematics at Basel in 1687, the same year that he published a significant article on geometry. He and Johann were both soon led into problems of infinitesimal geometry. The work of the brothers over the next several years was focused on the solution of puzzles that the leading mathematicians of Europe had proposed to demonstrate their own skill. Often a problem would be devised, solved by its formulator, and presented as a challenge to other mathematicians. One such problem was the shape of the curve that represented the motion of a body in constant descent in a gravitational field. Solutions would be offered, corrected by others, or counterproblems issued. This manner of solving problems greatly expanded the class of functions that could be analyzed using the tools of Leibnizian calculus. Jakob and Johann contributed to these sometimes-peevish arguments and mild polemics that nevertheless broadened the scope of calculus.

In 1695, Johann went to Groningen. He had no hope of getting the chair of mathematics at Basel, because his brother occupied it. The brothers were antagonistic toward each other. Jakob had taught Johann mathematics and apparently could never accept his younger brother as a professional equal. Both were sensitive, critical, and in need of recognition. Their intellectual gifts differed, as can best be seen by the famous problem of the brachistochrone posed by Johann in 1696. The brachistochrone is the curve a body makes as it moves along the path that takes the least time to travel between two points. Jakob solved the problem using a detailed but formally correct technique. Johann recognized that the problem could be rephrased in such a way that existing solutions could be adapted to the solution of this problem. Johann solved the problem in a more ingenious way, but Jakob recognized that his approach could be generalized. He laid the foundations of the field of the calculus of variations, which solves a wide variety of problems by using the methods of calculus to vary terms in an expression that takes a maximum or minimum. The brothers argued in print over each other’s solutions to another variational problem from 1696 to 1701.

Jakob I spent the remaining years of his life working on more problems and compiling the results of his life’s work. By the time of his death, he had accumulated a significant amount of original work on series (the finite sum of an infinite number of terms), gravitational theory, and engineering applications. Nikolaus I, Jakob’s nephew, helped to have Jakob’s most famous and original work, Ars conjectandi (1713; art of conjecture), published posthumously. Though incomplete, the work contains, among other things, Jakob’s final statements on probability theory. His contributions to probability theory are recognized to have been decisive in the further development of the field.

Johann broadened the scope of the new calculus in the mid-1690’s by calculating the details of the application of these methods to functions in which variables appear in the exponent. It was at this time that the brothers participated in their rancorous series of exchanges in print over the brachistochrone and other variational problems. Both brothers share the credit for the early development of the calculus of variations, for although Jakob first realized the generalizability of the technique, both followed up on this idea and applied it to other problems. After his brother’s death, Johann published several works that presented formal solutions to variational problems that were reminiscent of Jakob’s style.

Jakob’s death in 1705 was the cause of Johann’s return to Basel, where he took the vacant chair of mathematics. He became involved in the priority disputes between Sir Isaac Newton and Gottfried Wilhelm Leibniz over the invention of calculus and demonstrated the superiority of Leibniz’s notation in the solution of particular problems. After 1705, Johann worked primarily on theoretical and applied mechanics. He published Théorie de la maneuvre des vaisseaux (1714; theory of the movement of ships), dealing with navigational problems and ship design. He also won three prizes offered by scientific academies by espousing the Cartesian vortex theory to explain the motion of planets. He criticized some aspects of Cartesianism, but some scholars claim that his undisputed status and support for Descartes’s vortex theory delayed continental acceptance of Newtonian physics, which banishes such vortices in favor of forces.

Daniel Bernoulli obtained a position in the St. Petersburg Academy in 1725 and remained there until 1733. In 1727, Leonhard Euler joined him. His most productive years were spent in St. Petersburg. He wrote an original treatise on probability, a work on oscillations, and a draft of his most famous work, Hydrodynamica (1738; Hydrodynamics by Daniel Bernoulli, 1968). He returned to Basel to lecture in medicine but continued to publish in the areas that interested him most—mathematics and mechanics. His father, Johann, tried to establish priority for the founding of the field of hydrodynamics by plagiarizing his son’s original work and predating the publication. This is only the worst of many examples of the antagonism that Johann felt toward his son.

Daniel began lecturing on physiology, which was more to his liking than medicine, in 1743 and was offered the chair of physics in 1750. He lectured on physics until 1776, when he retired. His most important contributions center on his work in rational mechanics. He returned to probability theory in 1760 with his famous work on the effectiveness of the smallpox vaccine, arguing that the vaccine could extend the average lifespan by three years. He published a few more minor works on probability theory through 1776. Throughout his career, Daniel won ten prizes of the Paris Academy on topics involving astronomy, magnetism, navigation, and ship design.

Significance

The Bernoulli family was instrumental in developing many new fields of mathematics in the eighteenth century. They mastered the Leibnizian notation of calculus and successfully applied it to a range of problems. Their contributions to probability theory, the calculus of variation, differential and integral calculus, and the theories of series and of rational mechanics dominate even introductory textbooks in physics, mathematics, and engineering. The three Bernoullis who are the most famous of the eight who achieved contemporary recognition are Johann I, Jakob I, and Daniel.

Jakob was much more interested in mathematical formalism than his more intuitive younger brother Johann. Jakob’s main contribution was in the ingenious solutions to individual problems. The cumulative weight of these mounting solutions reflected on the power and scope of the newly emerging analytical techniques. He also contributed in important ways to probability theory, algebra, the calculus of variations, and the theory of series. Johann can claim similar contributions, for the brothers often worked on similar problems and criticized each other’s solutions in print. Johann also contributed to theoretical and applied mechanics. Daniel’s contributions include founding the field of hydrodynamics and making essential contributions to rational mechanics, probability theory, and the mechanics of physiology.

Other Bernoullis of note include Nikolaus I (1687-1759), Nikolaus II (1695-1726), Johann II (1710-1790), Johann III (1744-1807), and Jakob II (1759-1789). All received recognition from their contemporaries but did not make as many or as important contributions as their more famous relatives.

Bibliography

Bell, Eric T. Men of Mathematics. New York: Simon & Schuster, 1986. Addressed to the general reader, the book recounts the history leading to the major ideas of modern mathematics.

Brett, William F., Emile B. Feldman, and Michael Sentolwitz, eds. An Introduction to the History of Mathematics, Number Theory, and Operations Research. New York: MSS Information, 1974. Written for undergraduates, this book describes how mathematicians are creative and inquisitive people who sometimes find solutions that are useful but inaccurate.

Brown, Harcourt. “From London to Lapland: Maupertuis, Johann Bernoulli I, and La Terre applatic, 1728-1738.” In On Literature and History in the Age of Ideas: Essays on the French Enlightenment Presented to George R. Havens, edited by Charles G. S. Williams. Columbus: Ohio State University Press, 1975. Based on the voluminous collection of the Bernoullis’ letters, with a significant number of excerpts from the letters used to detail the controversy over the shape of the Earth.

Dunham, William. Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, 1990. This popular history of mathematics includes a chapter on the Bernoullis and the harmonic series.

‗‗‗‗‗‗‗. The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities. New York: Wiley & Sons, 1994. Includes a chapter on the Bernoulli trials.

James, Ioan. Remarkable Mathematicians: From Euler to von Neumann. New York: Cambridge University Press, 2002. The chapter on Euler explains how he was influenced by the work of the Bernoullis.

Lick, Dale W. “The Remarkable Bernoulli Family.” Mathematics Teacher 62 (May, 1969): 401-409. Presents brief biographical information on eight of the most prominent family members. Interweaves historical and technical aspects but presents only a few simple equations.

Simmons, George Finlay. Calculus Gems: Brief Lives and Memorable Mathematics. New York: McGraw-Hill, 1992. This collection of biographies includes a chapter on the Bernoullis’ contributions to mathematics.

Turnbull, H. W. The Great Mathematicians. 4th ed. New York: New York University Press, 1961. Aims at revealing the spirit of mathematics without burdening the reader with technical details. Chapter 8 deals with the Bernoullis and Euler.