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, structure, space, models, and change.
History
One ...
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 mathema ...
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
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. 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 (probability theory), 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 ...
, where he derived the first version of the
law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
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) Bernoulli](_blank)
The 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
). Swiss law does not designate a ''capital'' as such, but the federal parliament and government are installed in Bern, while other federal institutions, such as the federal courts, are in other cities (Bellinzona, Lausanne, Luzern, Neuchâtel ...
. 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
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
. 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 Continent#Subcontinents, subcontinent of Eurasia ...
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
Robert Hooke FRS (; 18 July 16353 March 1703) was an English polymath active as a scientist, natural philosopher and architect, who is credited to be one of two scientists to discover microorganisms in 1665 using a compound microscope that ...
. 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 ar ...
s.
![Acta Eruditorum - X astronomia, 1682 – BEIC 13349171](https://upload.wikimedia.org/wikipedia/commons/f/f4/Acta_Eruditorum_-_X_astronomia%2C_1682_%E2%80%93_BEIC_13349171.jpg)
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
Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of ...
's ''De ratiociniis in aleae ludo'',
Descartes' ''
La Géométrie
''La Géométrie'' was published in 1637 as an appendix to ''Discours de la méthode'' (''Discourse on the Method''), written by René Descartes. In the ''Discourse'', he presents his method for obtaining clarity on any subject. ''La Géométrie ...
'' and
Frans van Schooten's supplements of it. He also studied
Isaac Barrow
Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for proof of the fundamental theorem ...
and
John Wallis
John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...
, 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 L ...
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
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
, but he did show that it converged to a finite limit less than 2.
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
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
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
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 e ...
,
:
Jacob Bernoulli also discovered a general method to determine
evolutes
In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curv ...
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 mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
, 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"). Mor ...
and
epicycloids
In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an ''epicycle''—which rolls without slipping around a fixed circle. It is a particular kind of roulette.
Equat ...
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. I ...
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
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
, 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
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is c ...
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 ):
:
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 $1.00×1.25
4 = $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
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
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"). Mor ...
and the motto ''
Eadem mutata resurgo
''Eadem mutata resurgo'' is a Latin language, 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''.
...
'' ('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
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
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