Swiss mathematician Jakob Bernoulli (1654-1705) devoted his career to the study of calculating complex numerical formulas. Sometimes called Jacques or James Bernoulli to distinguish him from other prominent family members, he was the first in a long line of Bernoulli mathematicians that furthered the discipline greatly during the Age of Reason. He and his younger brother Johann engaged in a spirited, if not sometimes malicious, professional rivalry. Both, noted in an essay in DISCovering World History, "contributed to these sometimes-peevish arguments and mild polemics that nevertheless broadened the scope of calculus."

# Jakob Bernoulli

## Earned Theology Degree

Jakob Bernoulli was born in Basel, Switzerland, on January 6, 1654. The Bernoulli family was originally of Belgian origin and had been spice merchants in the Spanish Netherlands for sometime before 1583. In that year, Bernoulli's druggist grandfather moved the family to Basel, Switzerland, to escape after anti-Protestant religious persecution by a Roman Catholic ruling dynasty. In Basel, Bernoulli's father Nikolaus held civil posts on the town council and served as a magistrate after marrying a woman from a prominent banking family in the city.

As a young man, Bernoulli studied at the University of Basel, where he followed his father's wishes and earned a degree in philosophy in 1671 and then a theology degree five years later. Disinclined to enter the ministry, he studied mathematics and astronomy in his spare time. In 1676, he found work as a tutor in another Swiss city, Geneva, and then lived two years in France, during which time he studied the works of mathematician and philosopher René Descartes. Around this time, Bernoulli began his *Meditationes,* a scientific diary. He traveled to the Netherlands in 1681 and then on to England; as he had in France, he came to know prominent mathematicians of the day and began what would become a lifelong correspondence. Among them were Anglo-Irish physicist Robert Boyle, often referred to as the father of modern chemistry.

## Settled in Basel

Initially, Bernoulli was fascinated by the relationship of mathematics to the cosmos, and his first scientific papers dealt with gravity and the path of comets. He formulated a theory on the origins of comets, which later proved incorrect. Back in Basel by 1682, Bernoulli published his *De gravitate aetheris,* a treatise on the theory of gravity, which caused a stir in learned European circles at the time. In the city of his birth, he taught mechanics at the university after 1683 and conducted experiments in the field as well. He began publishing his findings in two top European journals of scientific study at the time, *Journal des sçavans* and *Acta eruditorum,* the latter published out of Leipzig, Germany. He was offered a position in the church at one point, but declined it. In 1684 he married Judith Stupanus, with whom he had a son and a daughter.

Bernoulli's pamphlet on parallels of logic and algebra appeared in 1685, again furthering his reputation, and in 1687 he was offered a post at the University of Basel as a professor of mathematics. He retained the job for the rest of his life. He began to delve into the works of Gottfried Wilhelm von Leibniz, a German philosopher and mathematician who enjoyed great prominence during this era. Primarily a philosopher known for his views on consistent rationalism, Leibniz was also an eminent mathematician who devised the first calculating machine, though it did not function well due to mechanical difficulties. Around 1676 Leibniz developed his fundamental principles of calculus, the branch of mathematics that investigates continuously
changing quantities. In a 1684 issue of *Acta eruditorum,* Leibniz published his theories on the differential calculus, which involves the study of the limit of a quotient as a denominator nears zero. Leibniz called this process "infinitesimal calculus," a term that denoted quantities smaller than any definable finite quantity, yet still larger than zero; the "infinitesimal" part was later abandoned.

## Influenced by Leibniz

Bernoulli worked studiously from Leibniz's findings. "Misunderstood by most of his colleagues, Leibniz's discovery nevertheless attracted a small following of mathematicians who realized the tremendous analytical power of calculus," asserted an essay in *Notable Mathematicians.* "Among Leibniz's followers, Bernoulli was among the first who completely grasped the essence of calculus, and he proceeded, in numerous contributions to Acta eruditorum, to develop the foundations of calculus." Bernoulli also worked on Leibniz's quandary over the isochronous curve, and he wrote an important treatise in 1687 on geometry that determined a way to divide any triangle into four equal parts with two perpendicular lines.

Bernoulli's long and illustrious career was marred by the rivalry with his younger brother, Johann, who also became an eminent mathematician in his day. Again, their father had strongly discouraged Johann's interest in this science, and the younger Bernoulli duly studied the more practical discipline of medicine at the University of Basel in the early 1680s; at the time, however, he also studied mathematics under his brother. Johann took a post as professor of mathematics at the University of Gröningen in the Netherlands in 1694, and the antagonism between the two intensified over the years. Their arguments usually centered around mathematical riddles that were then standard methods for scientists of the day to exchange ideas throughout Europe. A scholar would devise a problem, solve it, and then send it out for others to solve under deadline.

## Declared Superiority of His Work

In 1689, Bernoulli published his *Treatise on Infinite Series,* later known as the "Bernoulli inequality," but its proof had been conducted by Johann. "With uncharacteristic fraternal affection, Jakob even prefaced the argument with an acknowledgement of his brother's priority," wrote William Durham in his *Journey Through Genius: The Great Theorems of Mathematics.* The brothers also argued most vociferously over the riddle of the catenary curve, a term used to denote a line similar to a chain, fixed at two points, and hanging under its own weight. The riddle asked mathematicians to solve the equation for the curve. A generation before, the Italian astronomer, mathematician, and physicist Galileo had believed it to be a parabola, or a plane curve made up from all points equidistant from a given fixed point and a given fixed line. This proved false, and Bernoulli attempted to use calculus to solve it. He explored the possibilities in a 1690 paper, but his brother solved it before he finished. Johann delivered a wounding version of events, according to a source quoted in Durham's book. "The efforts of my brother were without success; for my part, I was more fortunate, for I found the skill (I say it without boasting, why should I conceal the truth?) to solve it in full… . It is true that it cost me study that robbed me of rest for an entire night … but the next morning, filled with joy, I ran to my brother, who was still struggling miserably with this Gordian knot without getting anywhere, always thinking, like Galileo, that the catenary was a parabola. Stop! Stop! I say to him, don't torture yourself any more to try and prove the identity of the catenary with the parabola, since it is entirely false."

Another famous quandary for the brothers Bernoulli involved the brachistochrone problem (from the Greek term meaning "shortest time"). In 1696, Johann published a riddle in *Acta Eruditorum* which invited others to determine the curve of quickest descent between two given points A and B, assuming that B does not lie right beneath A. Johann believed it was a cycloid curve, but his proof was wrong. "Jakob solved the problem using a detailed but formally correct technique," explained the essay in *DISCovering World History.* "Johann recognized that the problem could be rephrased in such a way that existing solutions could be adapted to the solution of this problem. Johann solved the problem in a more ingenious way, but Jakob recognized that his approach could be generalized."

The brothers disputed one another's works in the pages of leading scientific publications during their respective careers. Bernoulli, however, was more critical by nature and often entered into vituperative battles with his superiors at the University of Basel as well. "Each became the other's fiercest competitor in mathematical matters, until their attempts at one-upmanship seem, in retrospect, almost comical," noted Dunham in *Journey Through Genius.*

## Intrigued by Mollusk Shell

Bernoulli was fascinated by the mathematical properties of curves, especially the logarithmic spiral, a figure similar to the chambered nautilus mollusk shell in nature with its perfectly symmetrical spirals. It is also referred to as a *spira mirabilis,* or "wonderful spiral." As the essay in *Notable Mathematicians* explained, "Bernoulli noticed that the logarithmic spiral has several unique properties, including self-similarity, which means that any portion, if scaled up or down, is congruent to other parts of the curve." He was so intrigued by its shape that he requested his tombstone sport the motif, along with the Latin phrase *Eadem mutata resurgo* (Though changed, I arise again the same). Bernoulli died on August 16, 1705, in Basel, Switzerland. His brother succeeded him in his post as professor of mathematics at the University of Basel.

One of Bernoulli's best known works was published posthumously in 1713: *Ars conjectandi (The Art of Conjecture*), which involves probability theory. Described as "a highly innovative work" in an essay in *World of Scientific Discovery,* the tome "discussed what came to be known as the Bernoullian numbers and the Bernoulli theorem, and analyzed games of chance according to variations in players dexterity, expectation of profit, and other variables." It endured as an important tract on probability theory well into the modern age. A street in Paris's Eighth Arrondissement is
named in his honor. His nephew, Johann's son Daniel, formulated the Bernoulli principle, which involves the speed and properties of liquid or gas.

## Books

Dunham, William, *Journey Through Genius: The Great Theorems of Mathematics,* Wiley, 1990.

*Notable Mathematicians,* Gale Research, 1998.

Simonis, Doris, *Scientists, Mathematicians, and Inventors: Lives and Legacies,* Oryx, 1999.

*World of Scientific Discovery,* second edition, Gale, 1999.

## Online

*DISCovering World History,* Gale, 1997.

*U*X*L Science,* U*X*L, 1998.