Daniel Bernoulli

Swiss mathematician

• Born: February 8, 1700; Groningen, Netherlands
• Died: March 17, 1782; Basel, Switzerland

Eighteenth-century Swiss mathematician Daniel Bernoulli is best known for his work in the field of fluid dynamics, particularly the Bernoulli equation. His book Hydrodynamica gave the field its original name. Bernoulli also worked in the fields of physics and acoustics.

Primary field: Mathematics

Specialties: Mechanics; theoretical physics; acoustics

Early Life

Daniel Bernoulli (ber-NOO-lee) was born into a family of prominent Swiss mathematicians in Groningen, Netherlands, on February 8, 1700. His father, Johann Bernoulli, was a professor of mathematics at the University of Groningen, and his uncle Jakob Bernoulli was the chair of mathematics at the University of Basel, Switzerland. In 1705, when Bernoulli was five years old, his father took over his uncle’s position and the family moved back to Switzerland.

At the age of thirteen, Bernoulli was sent to the University of Basel to study philosophy and logic. He also excelled in mathematics and studied calculus under his older brother Nicolaus, their father having made significant discoveries in that discipline. At the age of sixteen, Bernoulli earned a master’s degree from Basel. Despite his skill and interest in mathematics, however, his father forbade him to pursue a career in the field. Johann Bernoulli tried at first to force his son to become a merchant, but Bernoulli refused. Johann then told his son to study to become a doctor. Bernoulli obliged, traveling to Italy to study medicine. He finished his studies in 1721. Unable to secure a teaching position, he continued to study medicine as well as mathematics.

While studying mathematics in Italy, Bernoulli wrote a treatise on probability and fluid motion. Published in Venice in 1724, Exercitationes quaedam mathematicae (Certain mathematial exercises) brought Bernoulli immediate recognition. He was offered a position as a professor of mathematics at the Saint Petersburg Academy of Sciences in Russia, where his brother Nicolaus also accepted an offer to teach mathematics. Before moving to Saint Petersburg in 1725, Bernoulli won first prize from the French Académie Royale des Sciences (Royal Academy of Sciences, now the French Academy of Sciences) for his essay on the best shape of hourglass to use on ships. It was the first of ten prizes he would win from the Academy.

Life’s Work

Bernoulli created the basis for his advances in mathematics, probability, and physics while teaching mathematics in Saint Petersburg. In addition to establishing him academically, his Exercitationes contained the origins of his exploration into fluid dynamics and probability. In 1726, he outlined the parallelogram of forces; the next year, he began regularly corresponding and collaborating with his friend Leonhard Euler, one of his father’s pupils.

While trying to learn more about the flow of blood with Euler, Bernoulli developed a way of measuring blood pressure. This involved sticking a tube in an artery and measuring the height at which the blood filled the tube. The method became so popular that it was used throughout Europe for approximately the next 170 years. Bernoulli’s method was later borrowed to measure airspeed.

Although Bernoulli found success in Russia, he was not happy there, and he left after eight years. Upon his return to Switzerland in 1733, he took a position at the University of Basel teaching botany, despite his lack of fondness for the subject. He continued working in other fields as well, such as mechanics and mathematics; in 1737, for example, he delivered a lecture on calculating the work done by the heart.

The next year, Bernoulli published his seminal Hydrodynamica (Hydrodynamics, 1738), establishing the field of hydrodynamics. This far-reaching work contained his famous fluid flow equation, called the Bernoulli equation, from which the Bernoulli principle was derived. The principle relates flow, speed, pressure, and potential energy. Hydrodynamica laid the foundation for all later work in hydrodynamics and aerodynamics, referred to collectively as fluid dynamics. Bernoulli devised a number of experiments to demonstrate his theories. He also examined gas pressure, positing that it was composed of fast and randomly moving particles. His analysis confirmed Robert Boyle’s 1660 gas law, which states that pressure multiplied by volume remains constant when the temperature does not change. This perspective paved the way for later studies, such as heat transfer.

Bernoulli also published a paper in 1738 that detailed the best shape for a ship’s anchor. The paper won a prize from the Royal Academy. That same year, he published “Specimen theoriae novae de mensura sortis” (“Exposition of a New Theory on the Measurement of Risk”), in which he investigated the Saint Petersburg paradox as a base for risk analysis and utility investigation. The Saint Petersburg paradox is a probability theory based on the Saint Petersburg gambling game, in which a player flips a coin until the head side appears. The winnings are two guilders (or two dollars) if the head appears on the first toss, four if on the second, eight if on the third, and so on ad infinitum. The probability of winning decreases by half for each flip of the coin: a 50 percent chance the first time, 25 percent the second time, and so on. The paradox is how much a player would be willing to pay to play the game. Bernoulli proposed that the solution was not to calculate the expected winnings but instead to calculate by a utility function, determining how useful the winnings would be in comparison to the player’s wealth.

In 1743, Bernoulli became a professor of physiology at the University of Basel. He used this opportunity to research subjects such as muscular contraction and the optic nerve. Seven years later, in 1750, he obtained the chair of experimental and speculative philosophy, now called theoretical physics. He was a very popular lecturer and continued to apply mathematics to physical phenomena.

Bernoulli was also elected a fellow of the Royal Society of London in 1750. Around this time, Euler and Bernoulli colaborated on the study of beam bending—that is, the sagging of a structure due to stress—and created a system later known as the Euler-Bernoulli beam equation. Their equation became the mathematical base for structural engineering projects such as the Eiffel Tower. Bernoulli also analyzed kinetic energy, which at the time was called vis viva, or living force. He posited that vis viva was conserved across the entire universe, anticipating the law of energy conservation, though he lacked the tools to prove it empirically.

As part of a scholarly dispute with Euler, Bernoulli investigated sound. He found that physical objects tend to vibrate at certain proper or natural frequencies. He named the lowest frequency the fundamental frequency and called the higher frequencies overtones. He also discovered that increases in frequency cause an increase in the number of nodes, or points, with no vibration. Bernoulli then built a mathematical framework around his findings, which were confirmed by Jean-Baptiste-Joseph Fourier’s work on harmonics in the early 1800s.

Much of Bernoulli’s later work involved the application of probability to disparate fields, including birth rate and inoculation. In one study in 1766, he used smallpox morbidity and mortality rates to illustrate the effectiveness of inoculation. In 1776, Bernoulli retired from teaching. He died in Basel, Switzerland, on March 17, 1782.

Impact

Bernoulli’s contributions to mathematics influenced numerous later developments, leading to improvements ranging from better sound quality in MP3s to stealthier submarines. He is considered the father of fluid dynamics, and his work on fluid flows is an integral part of the science used in the design of travel vessels, including airplanes, cars, and ships. Bernoulli’s fluid flow equation led to advances resulting in the modern practice of building ships based on model design, a process pioneered by naval architects such as William H. Froude (1810–79) and David Taylor (1864–1940), who then used fluid dynamics to predict the behavior of the full-size ship. Before these architects developed the idea of building ships in miniature first, the ships had to be built full scale before they could be tested.

Applied to aeronautics, Bernoulli’s principle of fluid dynamics was essential in the development of the first airfoils. An airfoil is the part of a travel vessel, particularly an airplane’s wing or propeller, that is designed to give the vessel speed in relation to the surrounding air pressure. The faster an airplane travels, the more lift it can achieve. Because Bernoulli’s approach worked, his equations were expanded upon, and they became the basis for a set of equations governing pressure. Initially used for low speeds, Bernoulli’s equations were extended to all velocity ranges, including modern hypersonic flight.

In addition to airplanes and submarines, Bernoulli’s equations became important to the automobile industry, enabling the production of faster and more fuel-efficient cars. Additionally, his application of probability to physics provided better definitions to temperature and other such fundamental ideas, allowing for more accurate descriptions and further work in the various fields of physics. For example, the field of thermodynamics, which studies the flow of heat as a group of excited particles, uses Bernoulli’s conjectures.

Bibliography

Baigrie, Brian S. The Renaissance and the Scientific Revolution: Biographical Portraits. New York: Scribner’s, 2001. Print. Contains a concise overview of Bernoulli and his work.

Bernoulli, Daniel, and Johann Bernoulli. Hydrodynamics and Hydraulics. Trans. Thomas Carmody and Helmut Kobus. Mineola: Dover, 2005. Print. A translation of Daniel Bernoulli’s treatise on fluids in motion, now called fluid dynamics, and Johann Bernoulli’s treatise on hydraulics.

Chakrabarti, Subrata K. The Theory and Practice of Hydrodynamics and Vibration. River Edge: World Scientific, 2002. Print. Covers the basics of fluid dynamics and the vibration of structures when subject to environmental loads. Applies this material to the design of structures.

Emrich, Raymond J. Fluid Dynamics. New York: Academic, 1981. Print. Focuses on the measure of velocity in fluid flows. Covers a variety of tracer and probe methods, as well as the analysis of Doppler shifts.

Šejnin, Oskar B. Portraits: Leonhard Euler, Daniel Bernoulli, Johann-Heinrich Lambert. Berlin: NG-Verl, 2009. Print. Provides biographical information about Euler, Bernoulli, and Lambert, translated from several languages. Discusses the close collaboration between Euler and Bernoulli and the impact Euler’s work had on the development of the Bernoulli equation.