Daniel Bernoulli

2.1. Family and Background

Daniel Bernoulli was born in Groningen, Netherlands, into the Bernoulli family, a lineage of highly distinguished mathematicians and scientists from Basel, Switzerland. The family's origins trace back to Antwerp, then part of the Spanish Netherlands, from where they emigrated to escape the Spanish persecution of Protestantism. After a brief period in Frankfurt, the family eventually settled in Basel. 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. He had two brothers, Niklaus and Johann II, both of whom also pursued careers in mathematics and physics. Daniel was widely regarded by contemporaries, including W. W. Rouse Ball, as "by far the ablest of the younger Bernoullis."

2.2. Childhood and Education

From an early age, Daniel showed a strong inclination towards mathematics and physics, despite his father's initial wishes. His father, Johann, initially encouraged Daniel to pursue a career in business, citing the poor financial compensation often associated with mathematicians. Daniel initially resisted, but later agreed to study business. His father then urged him to study medicine, to which Daniel consented on the condition that his father would provide him with private mathematics lessons.

Daniel pursued his medical studies at the University of Basel, Heidelberg, and the University of Strasbourg. In 1721, he earned a PhD in anatomy and botany, with his doctoral thesis focusing on the mechanics of respiration, applying the principle of conservation of energy. Even while studying medicine, he continued to learn calculus from his father and brother, demonstrating his unwavering interest in mathematics.

2.3. Relationship with Father

Daniel Bernoulli's relationship with his father, Johann Bernoulli, was notably complex and often strained, marked by intense academic competition and personal conflict. A significant incident occurred when both Daniel and Johann entered a prestigious scientific contest at the University of Paris and tied for first place. Johann, allegedly unable to tolerate the "shame" of his son being considered his equal, reportedly banned Daniel from his house.

The relationship further deteriorated due to an alleged act of plagiarism. Johann Bernoulli is accused of plagiarizing key ideas from Daniel's seminal work, Hydrodynamica, in his own book, Hydraulica. Johann reportedly backdated the publication of Hydraulica to appear as if it predated Daniel's Hydrodynamica, thereby attempting to claim originality for Daniel's discoveries. Despite Daniel's repeated attempts to reconcile with his father, Johann remained resentful until his death. This academic rivalry and personal animosity cast a long shadow over Daniel's early career and family life.

3.1. Professorships and Early Work

After completing his studies, Daniel Bernoulli initially struggled to secure a professorship at the University of Basel. He moved to Venice for practical medical training, where he continued his scientific pursuits. In 1724, with the assistance of Christian Goldbach, he published his earliest mathematical work, Exercitationes (Mathematical Exercises), which included solutions to differential equations proposed by Jacopo Riccati. The same year, he designed a maritime hourglass for navigation, which earned him the prestigious Paris Academy Grand Prize in 1725.

In 1725, Daniel, along with his brother Nicolaus II, accepted professorships in mathematics at the Saint Petersburg Academy of Sciences. However, Nicolaus II tragically died just eight months after their arrival. Daniel found his time in St. Petersburg challenging, experiencing unhappiness, a temporary illness, and disagreements over his salary, compounded by censorship from the Russian Orthodox Church. These factors provided him with an excuse to depart St. Petersburg in 1733.

Upon his return to Basel, he secured a professorship in botany at the University of Basel in 1734. He subsequently held chairs in medicine, metaphysics, and natural philosophy until his death. From 1750 to 1766, he held the coveted physics professorship, expanding his research interests to include astronomy and oceanography. Throughout his career, he was a prolific recipient of the Paris Academy Grand Prize, winning it a remarkable ten times.

3.2. Collaboration with Euler

Daniel Bernoulli maintained a lifelong and close friendship with Leonhard Euler, a fellow Swiss mathematician who was also a student of Daniel's father, Johann Bernoulli. Their scientific collaboration was particularly productive during their shared time at the Saint Petersburg Academy of Sciences, where Euler joined Daniel in 1727, a move facilitated by Johann Bernoulli. For six years, until Daniel's departure in 1733, they engaged in mutual assistance and high-level research.

Their collaborative efforts extended to various fields, including studies on elasticity and the development of the Euler-Bernoulli beam equation. They also jointly received a prize from the French Academy for their memoirs on the theory of tides, a work that, along with a contribution from Colin Maclaurin, encapsulated all significant advancements in the subject between Isaac Newton's Philosophiæ Naturalis Principia Mathematica and the later investigations of Pierre-Simon Laplace. Even after Daniel left St. Petersburg, and Euler later moved to Berlin, their intellectual exchange and friendship continued uninterrupted throughout their lives.

3.3. Major Works

Daniel Bernoulli's most significant work is Hydrodynamica, published in 1738. This seminal book laid the foundation for modern fluid mechanics and is celebrated for its systematic approach, where all results are derived from a single overarching principle, namely the conservation of energy. In this regard, it shares a conceptual similarity with Joseph Louis Lagrange's later Mécanique Analytique.

Within Hydrodynamica, Bernoulli articulated what is now known as Bernoulli's principle, which describes the inverse relationship between the velocity and pressure of a fluid. He also made early contributions to the kinetic theory of gases, using the concept of small particles to explain Boyle's law. The book also explored various mechanical questions, including the problems related to vibrating strings, building upon solutions proposed by Brook Taylor and Jean le Rond d'Alembert.

His earlier work, Exercitationes (Mathematical Exercises), published in 1724, showcased his early mathematical prowess. Two years later, in 1726, he was the first to emphasize the utility of resolving complex compound motions into simpler motions of translation and rotation.

4.1. Fluid Dynamics and Bernoulli's Principle

Bernoulli's most enduring contribution to physics is his foundational work in fluid dynamics, culminating in the formulation of Bernoulli's principle. This principle, detailed in his 1738 masterpiece Hydrodynamica, states that for an ideal fluid flowing along a streamline, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. Conversely, a decrease in fluid speed corresponds to an increase in pressure or potential energy. This principle essentially represents a specific application of the conservation of energy to fluid flow.

Bernoulli's principle has profound and widespread applications in various technologies and fields. It is critically important in aerodynamics, explaining how aircraft wings generate lift by creating differences in air pressure due to varying air speeds over their surfaces. It is also fundamental to the operation of carburetors, which use fluid dynamics to mix air and fuel for internal combustion engines, and De Laval nozzles, used in rocket engines and steam turbines.

Beyond these, Bernoulli's work on fluid dynamics also had early medical applications. He collaborated with Euler on studying the flow of liquids, particularly investigating the relationship between blood flow velocity and blood pressure. Bernoulli devised a method to measure blood pressure by inserting a straw-like tube into an artery; the height to which the blood rose in the straw indicated the pressure. This method, though invasive, was used by physicians across Europe for approximately 170 years until a less painful technique was developed in 1896. Interestingly, the same principle is still utilized today in aircraft to measure air speed. He also conceived the idea of propelling ships using recoil, an early precursor to modern jet propulsion.

4.2. Kinetic Theory of Gases and Other Contributions

In Hydrodynamica, Daniel Bernoulli also laid crucial groundwork for the kinetic theory of gases. He applied the concept of small, constantly moving particles to explain the empirical relationship between the pressure and volume of a gas, known as Boyle's law. This early formulation provided a microscopic explanation for macroscopic gas behavior.

His collaborative work with Leonhard Euler extended to the field of elasticity. Together, they developed the Euler-Bernoulli beam equation, a fundamental equation used in engineering to calculate the load-bearing capacity and deflection of beams under various forces. This equation remains a cornerstone in structural analysis and design.

Furthermore, Bernoulli made significant contributions to wave mechanics. In 1753, he was the first to articulate the principle of superposition, stating that "The general motion of a vibrating system is given by a superposition of its proper vibrations." This principle is fundamental to understanding wave phenomena and the behavior of vibrating systems, such as vibrating strings, a topic on which he wrote numerous papers, analyzing solutions proposed by Brook Taylor and Jean le Rond d'Alembert.

5.1. Utility Theory and the St. Petersburg Paradox

In his seminal 1738 paper, Specimen theoriae novae de mensura sortis (Exposition of a New Theory on the Measurement of Risk), Daniel Bernoulli offered an innovative solution to the St. Petersburg paradox. This paradox highlights a situation where, in a game of chance, the expected monetary value of participating is infinite, yet most rational individuals would only be willing to pay a small finite amount to play.

Bernoulli's solution introduced the key economic concepts of risk aversion, risk premium, and, most importantly, utility. He observed that when making decisions involving uncertainty, people do not always aim to maximize their potential monetary gain. Instead, they strive to maximize "utility," an economic term encompassing their personal satisfaction and benefit. Bernoulli realized that while there is a direct relationship between money gained and utility, this relationship exhibits diminishing returns as the amount of money gained increases. For example, an additional 100 USD in income would provide significantly more utility to a person earning 10.00 K USD per year than it would to someone earning 50.00 K USD per year.

He famously stated: "The satisfaction derived from an increase in wealth, however small, is inversely proportional to the amount of wealth previously possessed." He further noted, "Considering human nature, the aforementioned assumption seems valid for many people. This also means that many people can apply this comparative concept."

By shifting the focus from the expected monetary value to the expected utility, Bernoulli demonstrated that the expected utility of playing the St. Petersburg game would be a finite value, thus resolving the paradox. This groundbreaking insight laid crucial groundwork for modern economic theory, particularly expected utility theory and behavioral economics. His ideas on marginal utility, though independently established by William Stanley Jevons over a century later, were fully revived and integrated into mainstream economics with the publication of John von Neumann and Oskar Morgenstern's Theory of Games and Economic Behavior in 1944.

5.2. Statistical Analysis

Bernoulli also made early methodological contributions to statistical analysis. In 1766, he conducted one of the earliest attempts to analyze a statistical problem involving censored data. His work involved analyzing smallpox morbidity and mortality data to empirically demonstrate the efficacy of inoculation as a public health measure. This study was a pioneering example of applying rigorous statistical methods to address critical public health issues, showcasing his interdisciplinary approach to scientific inquiry.

6.1. Honors and Awards

Throughout his illustrious career, Daniel Bernoulli received significant recognition for his scientific achievements. He was elected a Fellow of the Royal Society in May 1750, a testament to his standing within the international scientific community. His innovative work was frequently honored by the Paris Academy of Sciences, where he won the prestigious Grand Prize a remarkable ten times. These prizes recognized his contributions across various fields, including his early work on the maritime hourglass (1725) and his joint prize with his father and Euler for their memoirs on tidal theory (1734).

Posthumously, his enduring impact has continued to be celebrated. In 2002, Daniel Bernoulli was inducted into the International Air & Space Hall of Fame at the San Diego Air & Space Museum, acknowledging the fundamental importance of Bernoulli's principle to the field of aerodynamics and aviation.

6.2. Historical Evaluation and Controversies

Historically, Daniel Bernoulli is recognized as one of the most brilliant minds of the 18th century. His scientific achievements are highly regarded for their depth, originality, and the rigorous application of mathematical principles to real-world phenomena. He effectively combined Isaac Newton's theories with Leibniz's calculus, powerfully utilizing the conservation of energy principle in his mechanical analyses, particularly in fluid dynamics. His work laid the groundwork for entire fields, including modern fluid mechanics, the kinetic theory of gases, and expected utility theory in economics.

Despite his immense contributions, his legacy is also marked by the complex and often strained dynamics within his own family. The intense rivalry with his father, Johann Bernoulli, and the alleged plagiarism of Daniel's work in Hydrodynamica by Johann's Hydraulica remain notable controversies. These personal conflicts highlight the competitive academic environment of the era but also underscore Daniel's resilience and independent scientific spirit in the face of familial animosity. His influence on subsequent scientific and economic thought is undeniable, with his ideas continuing to be foundational in various disciplines centuries after his passing.