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Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent
mathematicians A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
in the
Bernoulli family The Bernoulli family () of Basel was a patrician family, notable for having produced eight mathematically gifted academics who, among them, contributed substantially to the development of mathematics and physics during the early modern period. ...
. He was an early proponent of Leibnizian calculus and sided with
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of math ...
during the
Leibniz–Newton calculus controversy In the history of calculus, the calculus controversy (german: Prioritätsstreit, lit=priority dispute) was an argument between the mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had first invented calculus. The question was a ...
. He is known for his numerous contributions to
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, and along with his brother
Johann Johann, typically a male given name, is the German form of ''Iohannes'', which is the Latin form of the Greek name ''Iōánnēs'' (), itself derived from Hebrew name '' Yochanan'' () in turn from its extended form (), meaning "Yahweh is Gracious ...
, was one of the founders of the calculus of variations. He also discovered the fundamental mathematical constant . However, his most important contribution was in the field of
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
, where he derived the first version of the law of large numbers in his work ''
Ars Conjectandi (Latin for "The Art of Conjecturing") is a book on combinatorics and mathematical probability written by Jacob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, apa ...
''.Jacob (Jacques) BernoulliThe MacTutor History of Mathematics archive
School of Mathematics and Statistics,
University of St Andrews (Aien aristeuein) , motto_lang = grc , mottoeng = Ever to ExcelorEver to be the Best , established = , type = Public research university Ancient university , endowment ...
, UK.


Biography

Jacob Bernoulli was born in
Basel , french: link=no, Bâlois(e), it, Basilese , neighboring_municipalities= Allschwil (BL), Hégenheim (FR-68), Binningen (BL), Birsfelden (BL), Bottmingen (BL), Huningue (FR-68), Münchenstein (BL), Muttenz (BL), Reinach (BL), Riehen (BS ...
, Switzerland. Following his father's wish, he studied
theology Theology is the systematic study of the nature of the divine and, more broadly, of religious belief. It is taught as an academic discipline, typically in universities and seminaries. It occupies itself with the unique content of analyzing the ...
and entered the ministry. But contrary to the desires of his parents, he also studied mathematics and
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
. He traveled throughout
Europe Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a subcontinent of Eurasia and it is located entirel ...
from 1676 to 1682, learning about the latest discoveries in mathematics and the sciences under leading figures of the time. This included the work of
Johannes Hudde Johannes (van Waveren) Hudde (23 April 1628 – 15 April 1704) was a burgomaster (mayor) of Amsterdam between 1672 – 1703, a mathematician and governor of the Dutch East India Company. As a "burgemeester" of Amsterdam he ordered that t ...
,
Robert Boyle Robert Boyle (; 25 January 1627 – 31 December 1691) was an Anglo-Irish natural philosopher, chemist, physicist, alchemist and inventor. Boyle is largely regarded today as the first modern chemist, and therefore one of the founders of ...
, and Robert Hooke. During this time he also produced an incorrect theory of
comet A comet is an icy, small Solar System body that, when passing close to the Sun, warms and begins to release gases, a process that is called outgassing. This produces a visible atmosphere or coma, and sometimes also a tail. These phenomena ...
s. Bernoulli returned to Switzerland, and began teaching mechanics at the
University of Basel The University of Basel (Latin: ''Universitas Basiliensis'', German: ''Universität Basel'') is a university in Basel, Switzerland. Founded on 4 April 1460, it is Switzerland's oldest university and among the world's oldest surviving universit ...
from 1683. His doctoral dissertation ''Solutionem tergemini problematis'' was submitted in 1684. It appeared in print in 1687. In 1684 Bernoulli married Judith Stupanus; they had two children. During this decade, he also began a fertile research career. His travels allowed him to establish correspondence with many leading mathematicians and scientists of his era, which he maintained throughout his life. During this time, he studied the new discoveries in mathematics, including Christiaan Huygens's ''De ratiociniis in aleae ludo'', Descartes' '' La Géométrie'' and Frans van Schooten's supplements of it. He also studied Isaac Barrow and John Wallis, leading to his interest in infinitesimal geometry. Apart from these, it was between 1684 and 1689 that many of the results that were to make up ''
Ars Conjectandi (Latin for "The Art of Conjecturing") is a book on combinatorics and mathematical probability written by Jacob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, apa ...
'' were discovered. He was appointed professor of mathematics at the
University of Basel The University of Basel (Latin: ''Universitas Basiliensis'', German: ''Universität Basel'') is a university in Basel, Switzerland. Founded on 4 April 1460, it is Switzerland's oldest university and among the world's oldest surviving universit ...
in 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brother
Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Le ...
on mathematical topics. The two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the differential calculus in "
Nova Methodus pro Maximis et Minimis "Nova Methodus pro Maximis et Minimis" is the first published work on the subject of calculus. It was published by Gottfried Leibniz in the ''Acta Eruditorum'' in October 1684. It is considered to be the birth of infinitesimal calculus. Full tit ...
" published in ''
Acta Eruditorum (from Latin: ''Acts of the Erudite'') was the first scientific journal of the German-speaking lands of Europe, published from 1682 to 1782. History ''Acta Eruditorum'' was founded in 1682 in Leipzig by Otto Mencke, who became its first editor, ...
''. They also studied the publications of von Tschirnhaus. It must be understood that Leibniz's publications on the calculus were very obscure to mathematicians of that time and the Bernoullis were among the first to try to understand and apply Leibniz's theories. Jacob collaborated with his brother on various applications of calculus. However the atmosphere of collaboration between the two brothers turned into rivalry as Johann's own mathematical genius began to mature, with both of them attacking each other in print, and posing difficult mathematical challenges to test each other's skills. By 1697, the relationship had completely broken down. The lunar crater
Bernoulli Bernoulli can refer to: People *Bernoulli family of 17th and 18th century Swiss mathematicians: ** Daniel Bernoulli (1700–1782), developer of Bernoulli's principle **Jacob Bernoulli (1654–1705), also known as Jacques, after whom Bernoulli numbe ...
is also named after him jointly with his brother Johann.


Important works

Jacob Bernoulli's first important contributions were a pamphlet on the parallels of logic and algebra published in 1685, work on probability in 1685 and geometry in 1687. His geometry result gave a construction to divide any triangle into four equal parts with two perpendicular lines. By 1689 he had published important work on
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
and published his law of large numbers in probability theory. Jacob Bernoulli published five treatises on infinite series between 1682 and 1704. The first two of these contained many results, such as the fundamental result that \sum diverges, which Bernoulli believed were new but they had actually been proved by
Pietro Mengoli Pietro Mengoli (1626, Bologna – June 7, 1686, Bologna) was an Italian mathematician and clergyman from Bologna, where he studied with Bonaventura Cavalieri at the University of Bologna, and succeeded him in 1647. He remained as professor there ...
40 years earlier and was proved by Nicole Oresme in the 14th century already. Bernoulli could not find a closed form for \sum, but he did show that it converged to a finite limit less than 2. Euler was the first to find the limit of this series in 1737. Bernoulli also studied the exponential series which came out of examining compound interest. In May 1690 in a paper published in ''Acta Eruditorum'', Jacob Bernoulli showed that the problem of determining the isochrone is equivalent to solving a first-order nonlinear differential equation. The isochrone, or curve of constant descent, is the curve along which a particle will descend under gravity from any point to the bottom in exactly the same time, no matter what the starting point. It had been studied by Huygens in 1687 and Leibniz in 1689. After finding the differential equation, Bernoulli then solved it by what we now call
separation of variables In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
. Jacob Bernoulli's paper of 1690 is important for the history of calculus, since the term
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
appears for the first time with its integration meaning. In 1696 Bernoulli solved the equation, now called the
Bernoulli differential equation In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form : y'+ P(x)y = Q(x)y^n, where n is a real number. Some authors allow any real n, whereas others require that n not be 0 or 1. The ...
, : y' = p(x)y + q(x)y^n. Jacob Bernoulli also discovered a general method to determine evolutes of a curve as the envelope of its circles of curvature. He also investigated caustic curves and in particular he studied these associated curves of the
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
, the
logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...
and epicycloids around 1692. The
lemniscate of Bernoulli In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. ...
was first conceived by Jacob Bernoulli in 1694. In 1695 he investigated the drawbridge problem which seeks the curve required so that a weight sliding along the cable always keeps the drawbridge balanced. Bernoulli's most original work was ''
Ars Conjectandi (Latin for "The Art of Conjecturing") is a book on combinatorics and mathematical probability written by Jacob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, apa ...
'', published in Basel in 1713, eight years after his death. The work was incomplete at the time of his death but it is still a work of the greatest significance in the theory of probability. The book also covers other related subjects, including a review of combinatorics, in particular the work of van Schooten, Leibniz, and Prestet, as well as the use of
Bernoulli numbers In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
in a discussion of the exponential series. Inspired by Huygens' work, Bernoulli also gives many examples on how much one would expect to win playing various games of chance. The term Bernoulli trial resulted from this work. In the last part of the book, Bernoulli sketches many areas of
mathematical probability Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, including probability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; and the law of large numbers. Bernoulli was one of the most significant promoters of the formal methods of higher analysis. Astuteness and elegance are seldom found in his method of presentation and expression, but there is a maximum of integrity.


Discovery of the mathematical constant e

In 1683 Bernoulli discovered the constant e by studying a question about compound interest which required him to find the value of the following expression (which is in fact ): :\lim_ \left( 1 + \frac \right)^n One example is an account that starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.00×1.5² = $2.25. Compounding quarterly yields $1.00×1.254 = $2.4414..., and compounding monthly yields $1.00×(1.0833...)12 = $2.613035.... Bernoulli noticed that this sequence approaches a limit (the force of interest) for more and smaller compounding intervals. Compounding weekly yields $2.692597..., while compounding daily yields $2.714567..., just two cents more. Using as the number of compounding intervals, with interest of 100% / in each interval, the limit for large is the number that Euler later named ; with ''continuous'' compounding, the account value will reach $2.7182818.... More generally, an account that starts at $1, and yields (1+) dollars at compound interest, will yield dollars with continuous compounding.


Tombstone

Bernoulli wanted a
logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...
and the motto ''
Eadem mutata resurgo ''Eadem mutata resurgo'' is a Latin phrase that literally translates to "''Although changed, I arise the same''". Background The word-for-word translation of the phrase is :"Same having-changed I-rise". :''Eadem mutata resurgo''. The sense is bet ...
'' ('Although changed, I rise again the same') engraved on his tombstone. He wrote that the self-similar spiral "may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self." Bernoulli died in 1705, but an Archimedean spiral was engraved rather than a logarithmic one. Translation of Latin inscription: :Jacob Bernoulli, the incomparable mathematician. :Professor at the University of Basel For more than 18 years; :member of the Royal Academies of Paris and Berlin; famous for his writings. :Of a chronic illness, of sound mind to the end; :succumbed in the year of grace 1705, the 16th of August, at the age of 50 years and 7 months, awaiting the resurrection. :Judith Stupanus, :his wife for 20 years, :and his two children have erected a monument to the husband and father they miss so much.


Works

* (title roughly translates as "A new hypothesis for the system of comets".) * *''Ars conjectandi, opus posthumum'', Basileae, impensis Thurnisiorum Fratrum, 1713. * ** Bernoulli - De gravitate aetheris, 1683 - 1216514.jpg, ''De gravitate aetheris'', 1683 Bernoulli, Jakob – Opera, vol 1, 1744 – BEIC 12199963.jpg, ''Opera'', vol 1, 1744


Notes


References


Further reading

* *


External links

* * * * * * Gottfried Leibniz and Jakob Bernoull
Correspondence Regarding the Art of Conjecturing"
{{DEFAULTSORT:Bernoulli, Jakob 1655 births 1705 deaths 17th-century apocalypticists 17th-century Swiss mathematicians 18th-century apocalypticists 18th-century Latin-language writers 18th-century male writers 18th-century Swiss mathematicians
Jacob Jacob (; ; ar, يَعْقُوب, Yaʿqūb; gr, Ἰακώβ, Iakṓb), later given the name Israel, is regarded as a patriarch of the Israelites and is an important figure in Abrahamic religions, such as Judaism, Christianity, and Islam. J ...
Burials at Basel Münster Members of the French Academy of Sciences Number theorists Scientists from Basel-Stadt Probability theorists Swiss mathematicians