Jacob Bernoulli

• 1. Introduction

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  Jacob Bernoulli died on the 16th of August 1705, and was buried in Basel with a spiral engraved on his tombstone. There was just one problem: the stonemasons carved the wrong spiral. Bernoulli had wanted a logarithmic spiral, the self-similar curve he had studied so closely. Instead, they gave him an Archimedean one. He had even written out why the logarithmic spiral moved him, describing it as a symbol of fortitude in adversity and of the human body restored to its perfect self after death. The motto he chose was Eadem mutata resurgo: "Although changed, I rise again the same." The error on his grave stands as an odd coda to a life defined by precision.

  Bernoulli grew up the son and grandson of Protestant spice merchants in Basel, in what is now Switzerland. His father wanted him in the clergy. He studied theology and entered the ministry. Then he quietly went ahead and mastered mathematics and astronomy anyway. That dual life, official duty alongside private obsession, set the tone for everything that followed. What drove him across Europe in the 1670s? What was the rivalry with his own brother that grew so bitter both men attacked each other in print? And how did a question about compound interest lead Bernoulli to a number that mathematicians still use every day?

• 2. A Continent for a Classroom

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  Between 1676 and 1682, Bernoulli traveled throughout Europe, seeking out the leading figures of contemporary mathematics and science. He met the ideas, if not always the people, behind a generation of discovery. The names he encountered on that journey included Johannes Hudde, Robert Boyle, and Robert Hooke. Not everything he produced during those years held up. He developed a theory of comets that later proved incorrect. But the journey gave him something more durable: a correspondence network with leading mathematicians and scientists that he maintained for the rest of his life.

  When Bernoulli returned to Switzerland, he began teaching mechanics at the University of Basel in 1683. His doctoral dissertation, Solutionem tergemini problematis, was submitted in 1684 and appeared in print in 1687. That same year, 1684, he married Judith Stupanus, with whom he would have two children. The marriage and the dissertation arrived in the same year his research career was also beginning to take shape.

  The reading he pursued during those years was deliberately wide. He worked through Christiaan Huygens's De ratiociniis in aleae ludo, Descartes' La Geometrie, and Frans van Schooten's supplements to it. He also studied Isaac Barrow and John Wallis, which drew him toward infinitesimal geometry. Between 1684 and 1689, many of the results that would eventually fill his masterwork Ars Conjectandi began to take form.

• 3. The Calculus Wars

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  In 1684, Gottfried Wilhelm Leibniz published a paper on the differential calculus in the journal Acta Eruditorum, under the title Nova Methodus pro Maximis et Minimis. The paper was, by most accounts, extraordinarily difficult to follow. Bernoulli was among the first mathematicians anywhere to seriously attempt to understand and apply what Leibniz had written. He worked through it alongside his younger brother Johann, tutoring him on the material as he went.

  Bernoulli sided firmly with Leibniz in the bitter priority dispute that eventually broke out between Leibniz and Isaac Newton over who had invented calculus. His loyalty was not merely political. He made genuine contributions to Leibnizian calculus, including work that helped establish what would become the calculus of variations, a field he and Johann developed together.

  Collaboration, however, curdled. As Johann's own mathematical talent matured, the brothers began to compete rather than cooperate. They posed difficult problems to each other and attacked each other in print. By 1697, the relationship had completely broken down. A lunar crater now bears both their names jointly, a permanent pairing of two men who had come to regard each other as rivals.

• 4. Curves and Constants

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  In May 1690, in a paper published in Acta Eruditorum, Bernoulli showed that the problem of the isochrone, a curve along which a particle descends under gravity from any starting point in exactly the same time, was equivalent to solving a first-order nonlinear differential equation. He then solved it using what mathematicians now call separation of variables. That 1690 paper carries a particular distinction: it is the earliest known text in which the word "integral" appears with its modern mathematical meaning.

  Six years later, in 1696, he solved what became known as the Bernoulli differential equation. Around 1692, he had investigated caustic curves, studying them in connection with the parabola, the logarithmic spiral, and epicycloids. The lemniscate of Bernoulli, the figure-eight shaped curve, was first conceived by him in 1694. In 1695, he turned to the drawbridge problem: finding the curve a cable must follow so that a sliding weight always keeps the drawbridge in balance.

  He also studied infinite series across five treatises published between 1682 and 1704. One of his results, the demonstration that a particular series diverges, he believed to be new. It had in fact been proved by Pietro Mengoli roughly forty years earlier, and by Nicole Oresme in the 14th century. Bernoulli could not find a closed form for another series, though he proved it converged to a finite limit below 2. It fell to Leonhard Euler to find that limit in 1737.

• 5. The Constant in the Compound

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  In 1683, Bernoulli encountered a question about compound interest that led him somewhere unexpected. He was trying to find what happens to an account starting at one dollar, paying one hundred percent interest per year, as compounding grows more and more frequent. Compounded once at year's end, the account reaches two dollars. Compounded twice, it reaches $2.25. Compounded quarterly, roughly $2.44. Monthly, roughly $2.61. Weekly, roughly $2.69. Daily, roughly $2.71.

  Bernoulli noticed the sequence was approaching a fixed limit, converging as the intervals grew smaller. He identified that limit existed, but it was Leonhard Euler who later named it e and established its precise value of approximately 2.7182818. The number appears throughout mathematics, from calculus to probability to physics. Bernoulli's path to it was entirely practical, rooted in a banking calculation rather than an abstract search for constants.

• 6. Ars Conjectandi

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  Bernoulli's most significant work was not published until 1713, eight years after his death. Ars Conjectandi appeared in Basel, issued by the Thurnisiorum brothers, and it arrived unfinished. Despite that, the work reshaped the theory of probability.

  The book draws on Huygens' earlier writing on games of chance and extends it considerably, offering many examples of expected winnings in various games. It reviews combinatorics, drawing on the work of van Schooten, Leibniz, and Prestet. It introduces what are now called Bernoulli numbers in a discussion of the exponential series. The term Bernoulli trial, used today in statistics to describe a random experiment with exactly two outcomes, comes directly from this work.

  In its final section, the book sketches an ambitious map of mathematical probability: probability as a measurable degree of certainty, the difference between moral and mathematical expectation, the distinction between a priori and a posteriori probability, and the weighing of available arguments. At the center of it all sits the law of large numbers, the principle that as a trial is repeated more and more times, the observed frequency of an outcome converges toward its true probability. Bernoulli derived the first rigorous version of that law. His tombstone inscription called him "the incomparable mathematician" and noted he had held the professorship at Basel for more than eighteen years and was a member of the Royal Academies of Paris and Berlin. Those facts are carved in stone. The spiral beside them is the wrong one.