The Bernoullis (or as they are sometimes, and perhaps more correctly, called, the Bernouillis) were a family of Dutch origin, who were driven from Holland by the Spanish persecutions, and finally settled at Bâle in Switzerland. The first member of the family who obtained distinction in mathematics was James.
James Bernoulli
Jacob or James Bernoulli was born at Bâle on December 27, 1654; in 1687 he was appointed to a chair in mathematics in the university there; and occupied it until his death on August 16, 1705.
He was one of the earliest to realize how powerful as an instrument of analysis was the infinitesimal calculus, and he applied it to several problems, but did not himself invent any new processes. His great influence was uniformly and successfully exerted in favour of the use of the differential calculus, and his lessons on it, which were written in the form of two essays in 1691 and are published in the second volume of his works, shew how completely he had even then grasped the principles of the new analysis. These lectures, which contain the earliest use of the term integral, were the first published attempt to construct an integral calculus; for Leibnitz had treated each problem by itself, and had not laid down any general rules on the subject.
The most important discoveries of James Bernoulli were his solution of the problem to find an isochronous curve; his proof that the construction for the catenary which had been given by Leibnitz was correct, and his extension of this to strings of variable density and under a central force; his determination of the form taken by an elastic rod fixed at one end and acted on by a given force at the other, the elastica ; also of a flexible rectangular sheet with two sides fixed horizontally and filled with a heavy liquid, the lintearia ; and lastly, of a sail filled with wind, the velaria . In 1696 he offered a reward for the general solution of isoperimetrical figures, that is, of figures of a given species and given perimeter which shall include a maximum area: his own solution, published in 1701, is correct as far as it goes. In 1698 he published an essay on the differential calculus and its applications to geometry. He here investigated the chief properties of the equiangular spiral, and especially noticed the manner in which various curves deduced from it reproduced the original curve: struck by this fact he begged that, in imitation of Archimedes, and equiangular spiral should be engraved on his tombstone with the inscription eadem numero mutata resurgo . He also brought out in 1695 an edition of Descartes's Géometrie . In his Ars Conjectandi , published in 1713, he established the fundamental principles of the calculus of probabilities; in the course of the work he defined the numbers known by his name and explained their use, he also gave some theorems on finite differences. His higher lectures were mostly on the theory of series; these were published by Nicholas Bernoulli in 1713.
John Bernoulli
John Bernoulli, the brother of James Bernoulli, was born at Bâle on August 7, 1667, and died there on January 1, 1748. He occupied the chair of mathematics at Groningen from 1695 to 1705; and at Bâle, where he succeeded his brother, from 1705 to 1748. To all who did not acknowledge his merits in a manner commensurate with his own view of them he behaved most unjustly: as an illustration of his character it may be mentioned that he attempted to substitute for an incorrect solution of his own on the problem of isoperimetrical curves another stolen from his brother James, while he expelled his son Daniel from his house for obtaining a prize from the French Academy which he had expected to receive himself. He was, however, the most successful teacher of his age, and had the faculty of inspiring his pupils with almost as passionate a zeal for mathematics as he felt himself. The general adoption on the continent of the differential rather than the fluxional notation was largely due to his influence.
Leaving out of account his innumerable controversies, the chief discoveries of John Bernoulli were the exponential calculus, the treatment of trigonometry as a branch of analysis, the conditions for a geodesic, the determination of orthogonal trajectories, the solution of the brachistochrone, the statement that a ray of light pursues such a path that Σ μds is a minimum, and the enunciation of the principle of virtual work. I believe that he was the first to denote the accelerating effect of gravity by an algebraical sign g , and he thus arrived at the formula v 2 = 2 gh the same result would have been previously expressed by the proportion . The notation φ x to indicate a function of x was introduced by him in 1718, and displaced the notation X or ξ proposed by him in 1698; but the general adoption of symbols like f , F , φ, ψ, ... to represent functions, seems to be mainly due to Euler and Lagrange.
The Younger Bernoullis
Several members of the same family, but of a younger generation, enriched mathematics by their teaching and writings. The most important of these were the three sons of John; namely Nicholas, Daniel, and John the younger; and the two sons of John the Younger, who bore the names of John and James. To make the account complete I add here their respective dates. Nicholas Bernoulli, the eldest of the three sons of John, was born on Jan. 27, 1695, and was drowned at St. Petersburg, where he was professor, on July 26, 1726. Daniel Bernoulli, the scond son of John, was born on Feb. 9, 1700, and died on March 17, 1782; he was professor first at St. Petersburg and afterwards at Bâle, and shares with Euler the unique distinction of having gained the prize proposed annually by the French Academy no less than ten times. John Bernoulli, the younger, a brother of Nicholas and Daniel, was born on May 18, 1710, and died in 1790; he also was a professor at Bâle. He left two sons, John and James: of these, the former, who was born on Dec. 14, 1744, and died on July 10, 1807, was astronomer-royal, and director of mathematical studies at Berlin; while the latter, who was born on Oct. 17, 1759, and died in July 1789, was successively professor at Bâle, Verona, and St. Petersburg.
Daniel Bernoulli
Daniel Bernoulli, whose name I mentioned above, and who was by far the ablest of the younger Bernoullis, was a contemporary and intimate friend of Euler, whose works are mentioned in the next chapter. Daniel Bernoulli was born on Feb. 9, 1700, and died at Bâle, where he was professor of natural philosophy, on March 17, 1782. He went to St. Petersburg in 1724 as professor of mathematics, but the roughness of the social life was distasteful to him, and he was not sorry when a temporary illness in 1733 allowed him to plead his health as an excuse for leaving. He then returned to Bâle, and held successively chairs of medicine, metaphysics, and natural philosophy there.
His earliest mathematical work was the Exercitationes , published in 1724, which contains a solution of the differential equation proposed by Riccati. Two years later he pointed out for the first time the frequent desirability of resolving a compound motion into motions of translation and motions of rotation. His chief work is his Hydrodynamique , published in 1738; it resembles Lagrange's Méchanique analytique in being arranged so that all the results are consequences of a single principle, namely, in this case, the conservation of energy. This was followed by a memoir on the theory of the tides, to which, conjointly with the memoirs by Euler and Maclaurin, a prize was awarded by the French Academy: these three memoirs contain all that was done on this subject between the publication of Newton's Principia and the investigations of Laplace. Bernoulli also wrote a large number of papers on various mechanical questions, especially on problems connected with vibrating strings, and the solutions given by Taylor and by D'Alembert. He is the earliest writer who attempted to formulate a kinetic theory of gases, and he applied the idea to explain the law associated with the names of Boyle and Mariotte.
Background and Historical Context
The Bernoulli family, originally from the Netherlands, was forced to leave their homeland due to Spanish religious persecutions and eventually settled in Basel, Switzerland. This family became one of the most influential dynasties in the history of mathematics and science, spanning several generations. Their work laid the foundation for many modern mathematical concepts, especially in calculus, probability, and physics. The Bernoullis lived during a time when mathematics was rapidly evolving, with the development of calculus by Newton and Leibniz. They were among the first to apply and expand these new ideas, making significant contributions that shaped the future of science.
About the Authors
The most prominent members of the Bernoulli family include James (Jacob) Bernoulli, his brother John Bernoulli, and the younger generation such as Daniel Bernoulli. James Bernoulli was a pioneer in applying calculus to solve complex problems, while John Bernoulli was known for his teaching and further development of calculus notation and methods. Daniel Bernoulli, the most famous of the younger Bernoullis, made groundbreaking contributions to fluid dynamics and kinetic theory of gases. Their works were not only mathematical but also deeply connected to physics and natural philosophy, reflecting the interdisciplinary nature of scientific inquiry during the Enlightenment.
Detailed Explanation and Significance
The Bernoulli family’s work is foundational in many areas:
-
Calculus and Analysis: James Bernoulli was among the first to understand the power of infinitesimal calculus. He introduced the term "integral" and worked on constructing integral calculus, which is essential for understanding areas under curves and solving differential equations.
-
Probability Theory: In his book "Ars Conjectandi," James Bernoulli laid down the fundamental principles of probability, which are crucial for statistics, risk assessment, and decision-making.
-
Physics and Mechanics: Daniel Bernoulli’s "Hydrodynamique" introduced principles that explain fluid flow and energy conservation. His work on the kinetic theory of gases helped explain gas laws, which are fundamental in chemistry and physics.
-
Mathematical Notation: John Bernoulli contributed to the notation used in calculus, such as using φ(x) to denote functions, which is still in use today.
These contributions are not just historical facts; they form the backbone of many scientific and engineering disciplines.
Lessons and Inspirations for Students
Studying the Bernoulli family’s story and works offers several valuable lessons:
-
Persistence and Passion: The Bernoullis were deeply passionate about mathematics and science. Their dedication, despite personal and professional conflicts, shows the importance of perseverance in learning and discovery.
-
Interdisciplinary Thinking: Their work combined mathematics, physics, and philosophy, encouraging students to think broadly and connect different fields of knowledge.
-
Innovation and Application: They demonstrated how abstract mathematical ideas can be applied to solve real-world problems, inspiring students to seek practical uses for their learning.
-
Ethics and Collaboration: While some family members had conflicts, the overall legacy highlights the importance of sharing knowledge and working together to advance science.
How Students Can Apply These Insights
-
In Learning: Emulate the Bernoullis’ curiosity by exploring beyond textbooks. Try to understand the 'why' behind formulas and theories, and apply them to solve problems.
-
In Daily Life: Use logical thinking and problem-solving skills in everyday decisions. Understanding probability, for example, can help in making informed choices.
-
In Social Interactions: The story of the Bernoullis also teaches about the value of humility and respect in collaborations. Recognizing others’ contributions can lead to better teamwork.
-
Developing Positive Attitudes: Cultivate a mindset of lifelong learning and resilience. The Bernoullis faced challenges but continued to innovate, a great example for students facing academic or personal difficulties.
Encouraging the Spirit of the Bernoullis
To nurture the spirit of the Bernoulli family, students should:
-
Engage actively with challenging subjects like mathematics and science, seeing them as tools to understand the world.
-
Participate in discussions, debates, and collaborative projects to develop communication and teamwork skills.
-
Reflect on the ethical dimensions of scientific work, valuing honesty and integrity.
-
Explore historical stories of scientists to appreciate the human side of discovery, making learning more relatable and inspiring.
By studying the Bernoullis, students not only gain knowledge but also learn attitudes and skills that will serve them in many areas of life, from academics to personal growth and social relationships.
