Jacob Bernoulli (1654–1705)

Let's kick things off with Jacob Bernoulli, the first notable figure in this illustrious family during the late 17th century. Jacob, who introduced the mathematical constant e, was undeniably brilliant. His seminal work, "Ars Conjectandi," transformed the field of probability theory. Born as the eldest son of Niklaus Bernoulli, a spice merchant and alderman of Basel, Jacob's journey into mathematics began with the writings of René Descartes and Blaise Pascal. In 1687, he became a professor of mathematics at the University of Basel, engaging with leading mathematicians like Gottfried Leibniz and Christiaan Huygens.

He is particularly recognized for his contributions to probability and combinatorics, notably in his book "Ars Conjectandi," published posthumously in 1713. In this work, Jacob introduced foundational concepts such as expected value, the law of large numbers, and the Bernoulli distribution. His proofs of the binomial theorem for rational exponents and significant formulas involving binomial coefficients, including Pascal's triangle, are noteworthy.

Jacob also made significant strides in the calculus of variations, a mathematical discipline focused on optimizing functions. He was the first to address the brachistochrone problem, determining the path of quickest descent under gravity, and solved the isoperimetric problem, which identifies the curve that encloses the maximum area for a given length. The term "lemniscate," which describes a figure-eight curve, was coined by him as he explored its properties.

Reflecting on his legacy, Jacob once remarked, "I recognize the lion by his paw." While we may identify brilliance through intellect, was it sheer genius that defined Jacob, or was he simply leveraging the flourishing field of mathematics? He passed away from tuberculosis in 1705 at the age of 50, laid to rest in the cloister of the Münster of Basel, with an epitaph reading: "Eadem mutata resurgo" (I rise again, changed but the same).

Johann Bernoulli (1667–1748)

Next in line was Johann, Jacob's younger brother. Did he live in Jacob's shadow? Not at all; he flourished there. Although he was perceived as riding on Jacob's success, Johann carved his own path, particularly in calculus. He was known to challenge Leibniz to mathematical duels via correspondence. As the younger sibling, he studied both medicine and mathematics at the University of Basel and became a professor of mathematics in 1695.

Johann was a leading proponent of infinitesimal calculus, a discipline he learned from Leibniz. He applied calculus to various phenomena, including particle motion and the behavior of pendulums. His work also encompassed the calculus of variations, where he independently solved the brachistochrone problem and introduced the tautochrone problem, establishing the equation crucial for identifying extremals of a functional.

His contributions extended beyond calculus; Johann made significant advancements in number theory, geometry, and analysis. He proved that e is irrational and worked with harmonic series and exponential functions. Moreover, he introduced polar coordinates and explored the curves generated by rotating points.

A remarkable educator, Johann mentored future mathematical luminaries like Leonhard Euler and Pierre Louis Maupertuis, alongside his sons, Daniel and Johann II. Johann Bernoulli passed away in 1748 at the age of 80, resting beside Jacob with the epitaph: "Archimedes, Newton, and he."

In the video "The Bernoullis: When Math is the Family Business," we delve into the fascinating lives of the Bernoulli family and their contributions to mathematics.

Daniel Bernoulli (1700–1782)

Then came Daniel Bernoulli, Johann's son, renowned for the 'Bernoulli Principle.' His groundbreaking work in fluid dynamics was transformative. However, we must ask: was he merely building on the foundation laid by his forebears? Daniel's "Hydrodynamica" was more than a brilliant piece; it encapsulated a family legacy steeped in mathematical exploration.

Daniel studied medicine and mathematics at the University of Basel, earning his doctorate in 1721. He traveled extensively, collaborating with notable scientists like Leonhard Euler and Alexis Clairaut. By 1725, he was appointed as a professor of mathematics at the University of St. Petersburg and later returned to the University of Basel.

His most notable work, "Hydrodynamica," published in 1738, applied energy conservation principles to fluid flow, formulating the equation that describes the relationship between pressure, velocity, and height of fluids. He elucidated the Venturi effect, which describes the reduction of pressure and the increase in fluid velocity as it traverses a narrow passage. His research also included blood flow dynamics and projectile motion.

In addition to fluid dynamics, Daniel contributed to probability theory, introducing concepts like expected utility. He addressed the St. Petersburg paradox, a problem involving a game with infinite expected value but finite utility, and developed the standard deviation, a critical measure of data dispersion.

Daniel passed away in 1782 at the age of 82, interred alongside his father and uncle, with an epitaph stating: "He was the greatest of the Bernoullis."

Other Influential Bernoullis

Nicolaus Bernoulli, another significant member of the family, often goes unnoticed despite his substantial contributions to probability and statistics. His exchanges with Pierre Rémond de Montmort were intellectually stimulating and pivotal to the field.

As the 18th century progressed, the Bernoulli legacy continued through Johann II, Daniel II, and Johann III. However, one must ponder: were they innovating or simply iterating upon existing knowledge?

Ultimately, the success of the Bernoulli family prompts reflection: was it sheer genius, or were they simply in the right place at the right time? Did they significantly influence mathematical history, or did they merely capitalize on the scientific revolution? While their legacy can seem romanticized, it’s essential to consider that the Bernoullis were perhaps a product of their era—a convergence of opportunity, intellect, and familial rivalry.

In conclusion, regardless of whether they were geniuses or simply fortunate, the Bernoullis indelibly shaped the scientific community. Their narrative transcends mere formulas; it embodies ambition, rivalry, and a relentless quest for knowledge—perhaps the true essence of the Bernoulli story.

In the video "Who Discovered 'e' And Revolutionized Probability," we explore how the Bernoulli family influenced modern mathematics and the evolution of probability theory.

References:

• Bernoulli, Jakob I. "THE BERNOULLI FAMILY." The 17th and 18th Centuries: Dictionary of World Biography, Volume 4 4 (2013): 122.
• Eves, Howard. "Historically Speaking — : The Bernoulli Family." The Mathematics Teacher 59.3 (1966): 276–278.
• Senn, Stephen. "Bernoulli family." Encyclopedia of Statistics in Behavioral Science (2005).

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