Jacob Bernoulli

Born January 6, 1655, Basel, Switzerland

Died August 16, 1705 in Basel, Switzerland

Jacob Bernoulli

Jacob Bernoulli, a Swiss mathematician, was the first to use the word integral. He studied catenary lines, the curves of overhangs. He was an early adopter of polar coordinates and discovered isochrones.

Jacob Bernoulli's father, Nicolaus Bernoulli (1623 - 1708), inherited the spice business founded by his father in Basel, first in Amsterdam and then in Basel. The family, from Belgium, was a refugee fleeing persecution by the Spanish rulers in the Netherlands. King Philip of Spain sent the Duke of Alba to the Netherlands in 1567 to punish those who opposed Spanish rule, strengthen their faith in Roman Catholicism, and re-establish Philip's authority. Alba set up the Trouble Commission, a court that convicted more than 12,000 people, but most, such as the Protestant Bernoulli family, fled the country.

Nicolaus Bernoulli was an important citizen of Basel, a member of the town council and a magistrate. Jacob Bernoulli's mother also came from a prominent family of bankers and local councillors in Basel. Jacob Bernoulli was the younger brother John Bernoulli and uncle Daniel Bernoulli. He was forced by his parents to study philosophy and theology, which he hated, and graduated from the University of Basel in 1671 with a master's degree in philosophy and a theological license in 1676.

While pursuing his university degree at Jacob Bernoulli, he studied mathematics and astronomy against his parents' wishes. It is worth mentioning that this is a typical pattern for many Bernoulli families who study mathematics, although they face the pressure of careers in other fields. However Jacob Bernoulli was the first to go down this path, so for him it was quite different, because before Jacob Bernoulli there was no mathematical tradition in the family. Later members of the family must have been greatly influenced by the tradition of learning mathematics and mathematical physics.

In 1676, after taking his degree in theology, Bernoulli moved to Geneva, where he served as a tutor. He then traveled to France and spent two years studying with his followers. Descartes was now led by Malebranch. In 1681 Bernoulli traveled, where he met many mathematicians, including the Dutch Hudde. He continued to study with Europe's leading mathematicians and scientists, and he went to England, where he met Boyle and Hooke. At this time, he developed a keen interest in astronomy and wrote a work that gave a false theory of comets. As a result of his travels, Bernoulli began to correspond with many mathematicians for many years.

Jacob Bernoulli returned to Switzerland and from 1683 taught mechanics at the University of Basel, where he gave a series of important lectures on the mechanics of solids and liquids. Because his degree was theology, it was natural for him to turn to the church, but although he was appointed to the church, he refused. Bernoulli truly loves mathematics and theoretical physics, and it is these subjects that he teaches and researches. During this time, he studied the major mathematical works of his time, including Descartes' Geometry and additional material from van Schooten's Latin edition. Jacob Bernoulli also studied the work of Wallis and Barrow, which led to an interest in geometry of infinitesimals. Jacob began publishing articles in 1682 in the journal Acta Eruditorum founded in Leipzig. In 1684 Jacob Bernoulli married Judith Spanus. They will have two children, a son (grandfather's name is Nicholas) and a daughter. Unlike many members of the Bernoulli family, these children did not become mathematicians or physicists. One of the most important events in Jacob Bernoulli's mathematical research occurred when his younger brother John Bernoulli began to study mathematical subjects. John

His father told him to study medicine, but while he was studying the subject, he asked his brother Jacob to teach him mathematics. Jacob Bernoulli was appointed professor of mathematics in Basel in 1687, and the two brothers began to study calculus, as Leibniz proposed in his paper on differentiation in Nova Methodus pro Maximis et Minimis, itemque Tangentibus, published in the Journal of Polymath in 1684. They also studied von Tschirnhaus's publications. It must be understood that Leibniz's publication on calculus was very obscure to mathematicians of the time, and Bernoulli was the first to try to understand and apply it. Leibniz's theory.

Although both Jacob and John were committed to solving similar problems, their relationship quickly went from being one of the collaborators to one of the rivals. John Bernoulli's boasting was the first reason Jacob attacked him, and Jacob wrote that John was his student and that his only achievement was to repeat what his teacher had taught him. Of course, this is a very unfair statement. Jacob continued to attack his brother in shameful and unnecessary ways, especially after 1697. However, he did not reserve public criticism for his brother. He criticized the university authorities in Basel and again made publicly critical remarks, which, as one might expect, put him in trouble at the university. Jacob probably felt that John was the most powerful mathematician of the two, which hurt Jacob's nature and meant that he always had to feel that he had won praise from all sides. Hoffman wrote in [1] :-

Sensitivity, irritability, mutual enthusiasm for criticism, and exaggerated need for approval alienated the brothers, and Jacob was slower but deeper in their intelligence.

As this sentence implies, the brothers were equally at fault in the quarrel. John would have liked Jacob's professor of mathematics in Basel, and he was certainly unhappy that he had to move to Holland in 1695. This was another factor in the complete breakdown of the relationship in 1697.

Of course, the debate between the brothers about who can get the most recognition is a particularly silly argument because they both contributed to the most important mathematics. It's hard to say whether the competition pushed them to do greater things, or whether they might have achieved more if they had continued their initial collaboration. We will now examine some of the major contributions made by Jacob Bernoulli at an important stage in the development of mathematics following Leibniz's work on calculus.

Jacob Bernoulli's first important contribution was the similarity between logic and algebra in a pamphlet published in 1685, with working probabilities in 1685 and geometry in 1687. His geometric results give a structure that can divide any triangle into four equal parts with two vertical lines.

By 1689, he had published important works on infinite series and the law of large numbers in probability theory. Probability as an explanation of relative frequency is to say that if an experiment is repeated many times, then the relative frequency of the event is equal to the probability of the event occurring. The law of large numbers is a mathematical explanation of this result. Jacob Bernoulli published five papers on infinite series between 1682 and 1704. The first two of them contain many results, such as the basic result \sum (1/n) ∑ (1/n) divergence, bernoulli thinks it is new, but in fact they were proven by Mengoli 40 years ago. Bernoulli could not find the closed form \sum (1/n^{2}) ∑ ( 1 / n2) but he did show that it converged to a finite limit less than 2. Euler was the first to find the sum of this series in 1737. Bernoulli also studied the exponential series derived from compound interest studies. 1690

In a paper published in Acta Eruditorum in May, Jacob Bernoulli showed that the problem of determining isochrones is equivalent to solving first-order nonlinear differential equations. An isochrone, or constant descent curve, is the curve of a particle descending from any point at exactly the same time under gravity to the bottom, regardless of the starting point. Huygens studied in 1687 and Leibniz in 1689. After finding the differential equation, Bernoulli solved it with what we now call variable separation. Jacob Bernoulli's 1690 paper is important for the history of calculus because the word integral first appeared in the meaning of integral. In 1696 Bernoulli solved this equation, now known as the "Bernoulli Equation"

y' = p(x)y + q(x)y^{n}是′=p ( x ) y + q ( x ) yn

Hoffman describes this part of the work as:-

... Proof of Bernoulli's contributions to the mathematics of infinities in antiquity and contemporary times and his perseverance and analytical ability to deal with particularly relevant problems, even those of a mechanically dynamic nature.

Jacob Bernoulli also discovered a general method of identifying the involute line of a curve as the envelope of its round of curvature. He also studied caustic curves, especially the correlation curves of parabolas, logarithmic spirals, and outlines around 1692. Bernoulli's double button was first conceived by Jacob Bernoulli in 1694. In 1695, he investigated the problem of suspension bridge suspension bridges designed to require the weights of the curve to slide along the cable at all times. Jacob Bernoulli's most original work was Ars Conjectandi, published in Basel in 1713

Eight years after his death. At the time of his death, the work was not yet complete, but it remained the most important work in probability theory. In his book, Bernoulli reviews other people's work on probability, notably that of van Schooten, Leibniz, and Prestet. The Bernoulli number appears in the book's discussion of the Index series. Many examples are given of how much one expects to win when playing various games of chance. There are some interesting ideas about what probabilities really are [1]:-

... Probabilities as measurable degrees of certainty; necessity and contingency; moral and mathematical expectations; prior probabilities; expectations win when the player is divided according to dexterity; consideration of all available arguments, their valuations, and computable assessments; the law of large numbers...

In [1] Hoffman summarizes Jacob Bernoulli's contribution as follows:

Bernoulli greatly advanced algebra, infinitesimal calculus, variational calculus, mechanics, series theory, and probability theory. He is willful, stubborn, aggressive, vengeful, and troubled by inferiority, but he is convinced of his abilities. With these characteristics, he is bound to collide with his brother with the same similar personality. However, he exerted the most lasting influence on the latter.

Bernoulli is one of the most important enablers of a formal approach to advanced analytics. In his approach to expression and expression, agility and elegance are rarely found, but there is maximum integrity.

Jacob Bernoulli continued to be president of the Mathematical Academy of Basel until his death in 1705, when the chairman was held by his brother Johann. Jacob had always felt that the properties of the logarithmic spiral were almost magical, and he demanded that the words "Endem Mutata Resurgo" in Latin be inscribed on his tombstone, meaning "I will appear the same, albeit changed".