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Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli
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Daniel Bernoulli FRS (/bɜːrˈnuːli/ bur-NOO-lee; Swiss Standard German: [ˈdaːni̯eːl bɛrˈnʊli];[1] 8 February [O.S. 29 January] 1700 – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics.[2] His name is commemorated in the Bernoulli's principle, a particular example of the conservation of energy, which describes the mathematics of the mechanism underlying the operation of two important technologies of the 20th century: the carburetor and the aeroplane wing.[3][4]

Key Information

Early life

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Frontpage of Hydrodynamica (1738)

Daniel Bernoulli was born in Groningen, in the Netherlands, into a family of distinguished mathematicians.[5] The Bernoulli family came originally from Antwerp, at that time in the Spanish Netherlands, but emigrated to escape the Spanish persecution of the Protestants. After a brief period in Frankfurt the family moved to Basel, in Switzerland.

Daniel was the son of Johann Bernoulli (one of the early developers of calculus) and a nephew of Jacob Bernoulli (an early researcher in probability theory and the discoverer of the mathematical constant e).[5] He had two brothers, Niklaus and Johann II. Daniel Bernoulli was described by W. W. Rouse Ball as "by far the ablest of the younger Bernoullis".[6]

He is said to have had a bad relationship with his father. Both of them entered and tied for first place in a scientific contest at the University of Paris. Johann banned Daniel from his house, allegedly being unable to bear the "shame" of Daniel being considered his equal. Johann allegedly plagiarized key ideas from Daniel's book Hydrodynamica in his book Hydraulica and backdated them to before Hydrodynamica.[citation needed] Daniel's attempts at reconciliation with his father were unsuccessful.[7]

When he was in school, Johann encouraged Daniel to study business citing poor financial compensation for mathematicians. Daniel initially refused but later relented and studied both business and medicine at his father's behest under the condition that his father would teach him mathematics privately.[7] Daniel studied medicine at Basel, Heidelberg, and Strasbourg, and earned a PhD in anatomy and botany in 1721.

He was a contemporary and close friend of Leonhard Euler.[8][9] He went to St. Petersburg in 1724 as professor of mathematics, but was very unhappy there. A temporary illness[7] together with the censorship by the Russian Orthodox Church[10] and disagreements over his salary gave him an excuse for leaving St. Petersburg in 1733.[11] He returned to the University of Basel, where he successively held the chairs of medicine, metaphysics, and natural philosophy until his death.[12]

In May 1750 he was elected a Fellow of the Royal Society.[13]

Mathematical work

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Daniel Bernoulli

His earliest mathematical work was the Exercitationes (Mathematical Exercises), published in 1724 with the help of Goldbach. Two years later he pointed out for the first time the frequent desirability of resolving a compound motion into motions of translation and motion of rotation. In 1729, he published a polynomial root-finding algorithm which became known as Bernoulli's method.[14] His chief work is Hydrodynamica, published in 1738. It resembles Joseph Louis Lagrange's Mécanique Analytique in being arranged so that all the results are consequences of a single principle, namely, the conservation of vis viva, an early version of 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 Colin Maclaurin, a prize was awarded by the French Academy: these three memoirs contain all that was done on this subject between the publication of Isaac Newton's Philosophiae Naturalis Principia Mathematica and the investigations of Pierre-Simon 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 Brook Taylor and by Jean le Rond d'Alembert.[6]

  • Outer spine of Bernoulli's Hydrodynamica
    A 1738 copy of Bernoulli's Hydrodynamica
  • First page of the first section of a 1738 copy of Hydrodynamica

Economics and statistics

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In his 1738 book Specimen theoriae novae de mensura sortis (Exposition of a New Theory on the Measurement of Risk),[15] Bernoulli offered a solution to the St. Petersburg paradox as the basis of the economic theory of risk aversion, risk premium, and utility.[16] Bernoulli often noticed that when making decisions that involved some uncertainty, people did not always try to maximize their possible monetary gain, but rather tried to maximize "utility", an economic term encompassing their personal satisfaction and benefit. Bernoulli realized that for humans, there is a direct relationship between money gained and utility, but that it diminishes as the money gained increases. For example, to a person whose income is $10,000 per year, an additional $100 in income will provide more utility than it would to a person whose income is $50,000 per year.[17]

One of the earliest attempts to analyze a statistical problem involving censored data was Bernoulli's 1766 analysis of smallpox morbidity and mortality data to demonstrate the efficacy of inoculation.[18]

Physics

[edit]

In Hydrodynamica (1738) he laid the basis for the kinetic theory of gases, and applied the idea to explain Boyle's law.[6]

He worked with Euler on elasticity and the development of the Euler–Bernoulli beam equation.[19] Bernoulli's principle is of critical use in hydrodynamics.

According to Léon Brillouin, the principle of superposition was first stated by Daniel Bernoulli in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations."[20]

Works

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Pieces qui ont remporté le Prix double de l'Academie royale des sciences en 1737
  • Pieces qui ont remporté le Prix double de l'Academie royale des sciences en 1737 (in French). Paris: Imprimerie Royale. 1737.

Legacy

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In 2002, Bernoulli was inducted into the International Air & Space Hall of Fame at the San Diego Air & Space Museum.[21]

See also

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References

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[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Daniel Bernoulli

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from Grokipedia
Daniel Bernoulli (8 February 1700 – 17 March 1782) was a Dutch-born Swiss mathematician and physicist renowned for his foundational work in fluid dynamics, particularly Bernoulli's principle, which states that an increase in the speed of a fluid results in a simultaneous decrease in its pressure or potential energy. A member of the illustrious Bernoulli family of mathematicians, he was the second son of Johann Bernoulli, a leading figure in early calculus, and nephew to Jacob Bernoulli, who pioneered probability theory. Bernoulli's diverse contributions spanned hydrodynamics, probability, and economics, including his resolution of the [[St. Petersburg paradox]] through the introduction of expected utility theory, which laid groundwork for modern decision theory under risk. Born in Groningen, Netherlands, while his father held a professorship there, Bernoulli initially pursued studies in logic and philosophy at the University of Basel, earning a bachelor's degree at age 15 and a master's at 16, before turning to medicine in Heidelberg, Germany, and Strasbourg, France, at his father's insistence. Despite familial pressures to avoid mathematics due to intense competition among the Bernoullis, he gravitated toward it, working briefly in Venice and securing an early triumph by winning the Paris Academy of Sciences' grand prize in 1725 at age 25 for an innovative hourglass design that minimized sand flow variability. In 1725, he accepted the chair of mathematics at the Imperial Academy in St. Petersburg, Russia, invited by Empress Catherine I, though he later returned to Europe amid professional rivalries with his father. Bernoulli's career peaked in Basel, Switzerland, where he returned in 1733 and was appointed to the chair of botany in 1734, which he exchanged for physiology in 1743 before succeeding to the chair of physics in 1750, while maintaining active involvement in prestigious academies, including the Royal Society in London, the Paris Academy of Sciences, the Berlin Academy, and the St. Petersburg Academy. His seminal 1738 publication, Hydrodynamica, systematically explored fluid motion, pressure, density, and viscosity, deriving principles that explain phenomena like the lift generated by airplane wings. Over his lifetime, he amassed an extraordinary 10 prizes from the Paris Academy between 1725 and 1749 for advancements in astronomy, gravity, and tides, cementing his legacy as one of the 18th century's most influential scientists.

Biography

Early life and family background

Daniel Bernoulli was born on February 8, 1700, in Groningen, Netherlands, to Johann Bernoulli, a prominent mathematician, and Dorothea Falkner. The Bernoulli family formed a renowned dynasty of scholars, with Daniel's uncles Jakob Bernoulli and Nicolaus I Bernoulli making significant contributions to mathematics, including advancements in calculus and probability theory. His brothers, the older Nicolaus II Bernoulli and younger Johann II Bernoulli, also pursued distinguished careers in mathematics and science, perpetuating the family's intellectual legacy across generations. In 1705, the family relocated to Basel, Switzerland, when Johann Bernoulli assumed the chair of mathematics at the University of Basel following Jakob's death. From an early age, Daniel received tutoring in mathematics from his father, mastering calculus by the age of 11 despite Johann's initial discouragement of such pursuits in favor of more practical fields like medicine or commerce, which he believed offered better financial prospects.

Education and early influences

Daniel Bernoulli began his formal education at the University of Basel in 1713, at the age of 13, initially focusing on philosophy and logic as per his father's wishes, though he earned his baccalaureate in 1715 and master's degree in 1716. Despite this curriculum, Bernoulli's interest in mathematics grew during this period, influenced by his family's scholarly background that provided access to advanced texts. Under pressure from his father, Johann Bernoulli, he shifted toward medicine, beginning specialized studies in that field after completing his early degrees. In pursuit of medical training, Bernoulli traveled to Heidelberg in 1718 and Strasbourg in 1719, where he immersed himself in anatomical and physiological studies, before returning to Basel to finalize his doctorate. He received his MD from the University of Basel in 1721, with a dissertation titled De motu musculari (On the Movement of the Muscles), which explored biomechanical principles underlying muscle function. Concurrently, Bernoulli pursued self-directed studies in mathematics, delving into Isaac Newton's works on fluxions and engaging with the broader intellectual circle surrounding his father, including figures from Johann Bernoulli's academic network and rivalries. Following his graduation, Bernoulli undertook brief travels across Europe from 1724 to 1725, including a stay in Venice where he practiced medicine and continued independent studies in mathematics and related sciences. During this time, he produced his first notable publication: a 1724 paper on infinite series published in Acta Eruditorum, demonstrating his emerging mathematical talent through rigorous analysis of convergence properties. These early experiences solidified Bernoulli's interdisciplinary foundation, blending medical knowledge with mathematical rigor under the guidance of familial and academic influences.

Academic career and positions

In 1725, Daniel Bernoulli was appointed professor of mathematics at the newly established Russian Academy of Sciences in St. Petersburg, an invitation extended by the academy's leadership following the death of its founder, Peter the Great. He held this position until 1733, during which he lectured on mathematics and mechanics while contributing to the institution's early development alongside his brother Nicolaus II. His brother Nicolaus II died of fever in November 1726, deeply affecting Daniel, who nonetheless continued his work at the academy until 1733. Upon returning to Basel in 1733, Bernoulli was appointed professor of botany and anatomy at the University of Basel, a role influenced by his father Johann Bernoulli, who advocated for his son's placement despite initial opposition to more prestigious mathematical chairs. In 1743, he transitioned to the chair of physiology, continuing until 1750, when he was promoted to the chair of physics—a position that enabled greater emphasis on natural philosophy and which he occupied until his retirement in 1776. Bernoulli's professional ascent was marked by tensions with his father, including disputes over academic succession at Basel and a bitter rivalry exacerbated by their joint award of the Paris Academy of Sciences' Grand Prize in 1734 for work on the application of mathematics to astronomy; Johann publicly accused Daniel of appropriating his ideas, resulting in a lifelong estrangement. Bernoulli's growing recognition included election as a foreign member of the Paris Academy around this period, as well as memberships in the Royal Society of London (elected 1750), the Prussian Academy of Sciences in Berlin, the Bologna Academy, and others. Early accolades featured the 1725 Grand Prize from the Paris Academy for his design of an hourglass that ensured a constant trickle of sand despite ship movements, underscoring his rising European stature.

Later years and death

In 1750, Daniel Bernoulli was appointed professor of physics at the University of Basel, a role he fulfilled with dedication until his retirement in 1776, during which he continued to lecture and conduct research despite the challenges of advancing age. His productivity remained notable, as evidenced by his winning the Paris Academy's Grand Prize ten times between 1725 and 1749 for works on topics ranging from astronomy to navigation. Bernoulli never married and had no children, channeling his energies fully into his academic and scientific endeavors rather than personal family life. Relations within the Bernoulli family, however, were marked by persistent tensions, stemming largely from a bitter dispute with his father, Johann Bernoulli, over shared credit for a 1734 Paris Academy prize; this led to Johann disowning Daniel and barring him from the family home, a rift that endured until Johann's death in 1748. Similar rivalries over academic recognition and inheritance lingered among his brothers and nephews, reflecting the competitive dynamics of the Bernoulli dynasty. Daniel Bernoulli died peacefully in his sleep on March 17, 1782, in Basel, Switzerland, at the age of 82. He was buried in the cloister of Basel Minster (Basel Cathedral). Although he left no direct descendants, the Bernoulli family's intellectual legacy persisted through his siblings' lines, with nephews such as Johann II and Jacob II, and later grandnephews, upholding the tradition of contributions to mathematics and science at institutions like the University of Basel.

Mathematical Contributions

Work in calculus and geometry

Daniel Bernoulli made important early contributions to the calculus of variations in the 18th century, building upon the foundational work of his father Johann and uncle Jacob on optimization problems. While Johann Bernoulli solved the brachistochrone problem in 1696—determining the curve of fastest descent under gravity—Daniel extended these techniques to more general isoperimetric problems, which involve finding curves that extremize a quantity subject to a fixed perimeter constraint. His approaches in the early 1700s emphasized direct methods for solving such variational problems, influencing later developments in continuous optimization. A notable example of Daniel Bernoulli's work in this area is his 1742 proposal of variational principles for the elastica problem, which seeks the equilibrium shape of a thin elastic rod under bending. In a letter to Leonhard Euler, Bernoulli suggested minimizing the integral of the square of the curvature along the curve, formulating it as ∫κ2 ds,\int \kappa^2 \, ds, where κ\kappa is the curvature and dsds is the arc length element. This variational setup, although not fully solved by Bernoulli at the time, marked a significant step toward systematic methods for elastic curves and inspired Euler's subsequent rigorous treatment. In geometry, Bernoulli addressed problems related to arc length and rectifiability of curves. In his later writings, he characterized classes of smooth curves whose total arc length between endpoints could be expressed using elementary functions, such as algebraic or transcendental expressions without integrals. His main theorem in this domain provides necessary and sufficient conditions for such rectifiability, distinguishing curves amenable to exact computation from those requiring transcendental methods; for instance, he showed that certain transcendental curves like the tractrix satisfy these properties while others do not. This work advanced the understanding of integrable geometric quantities and had implications for applications in mechanics. Bernoulli also contributed to techniques in integral calculus for geometric solids. During the 1720s, while in St. Petersburg, he developed methods for the quadrature—the computation of areas and volumes—of surfaces and solids of revolution, applying integration to rotate plane curves around axes and derive formulas for generated surfaces. These efforts built on Leibnizian calculus, providing explicit integrals for cases like the rotation of conic sections, such as S=2π∫aby1+(y′)2 dxS = 2\pi \int_a^b y \sqrt{1 + (y')^2} \, dx
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