Editing Jacob Bernoulli

{{short description|Swiss mathematician (1655–1705)}}
{{for|other family members named ''Jacob''|Bernoulli family}}
{{Infobox scientist
| name              = Jacob Bernoulli
| image             = Jakob_Bernoulli.jpg
| caption           =
| birth_date        = {{birth date|df=yes|1655|1|6}}
| birth_place       = [[Basel]], [[Old Swiss Confederacy|Switzerland]]
| death_date        = {{death date and age|df=yes|1705|8|16|1655|1|6}}
| death_place       = Basel, Switzerland
| field             = [[Mathematics]], [[mechanics]]
| work_institution  = [[University of Basel]]
| education         = [[University of Basel]]<br>(D.Th., 1676; Dr. phil. hab., 1684)
| thesis1_title     = Primi et Secundi Adami Collatio
| thesis1_url       = 
| thesis1_year      = 1676
| thesis2_title     = Solutionem tergemini problematis arithmetici, geometrici et astronomici... (Solution to a triple problem in arithmetics, geometry and astronomy...)
| thesis2_url       = 
| thesis2_year      = 1684
| doctoral_advisor  = [[Peter Werenfels]]<br>(1676 thesis advisor)
| academic_advisors = [[Gottfried Wilhelm Leibniz]] (epistolary correspondent)
| doctoral_students = [[Jacob Hermann (mathematician)|Jacob Hermann]]<br>[[Nicolaus I Bernoulli]]
| notable_students  = [[Johann Bernoulli]]
| known_for         = [[Bernoulli differential equation]]<br>[[Bernoulli numbers]]<br>[[Bernoulli's formula]]<br>[[Bernoulli polynomials]]<br>[[Bernoulli map]]<br>[[Bernoulli trial]]<br>[[Bernoulli process]]<br>[[Bernoulli scheme]]<br>[[Bernoulli operator]]<br>[[Hidden Bernoulli model]]<br>[[Bernoulli sampling]]<br>[[Bernoulli distribution]]<br>[[Bernoulli random variable]]<br>[[Bernoulli's Golden Theorem]]<br>[[Bernoulli's inequality]]<br>[[Lemniscate of Bernoulli]]<br>[[Bernoulli's triangle]]
| footnotes         = Brother of [[Johann Bernoulli]] 
}}

'''Jacob Bernoulli'''{{efn|{{IPAc-en|lang|b|ɜːr|ˈ|n|uː|l|i}} {{respell|bur|NOO|lee}};<ref>{{cite LPD|3}}</ref> {{IPA|de-CH|ˈjaːkɔb bɛrˈnʊli|lang}}.<ref>Mangold, Max (1990). ''Duden — Das Aussprachewörterbuch''. 3. Auflage. Mannheim/Wien/Zürich, Dudenverlag.</ref>}} (also known as '''James''' in English or '''Jacques''' in French; {{OldStyleDateDY|6 January|1655|27 December 1654}} – 16 August 1705) was a Swiss mathematician. He sided with [[Gottfried Wilhelm Leibniz]] during the [[Leibniz–Newton calculus controversy]] and was an early proponent of Leibnizian [[calculus]], to which he made numerous contributions. A member of the [[Bernoulli family]], he, along with his brother [[Johann Bernoulli|Johann]], was one of the founders of the [[calculus of variations]]. He also discovered the fundamental mathematical constant {{mvar|[[e (mathematical constant)|e]]}}. However, his most important contribution was in the field of [[probability]], where he derived the first version of the [[law of large numbers]] in his work ''[[Ars Conjectandi]]''.<ref name="MacTutor">[http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Jacob.html Jacob (Jacques) Bernoulli], [http://www-gap.dcs.st-and.ac.uk/~history/ The MacTutor History of Mathematics archive], School of Mathematics and Statistics, [[University of St Andrews]], UK.</ref>

==Biography==
Jacob Bernoulli was born in [[Basel]] in the [[Old Swiss Confederacy|Swiss Confederation]],<ref>{{cite book|last1=Sensenbaugh |first1=Robert|date=13 September 2013|chapter=THE BERNOULLI FAMILY|chapter-url=https://books.google.co.uk/books?hl=en&lr=&id=HaHdAAAAQBAJ&oi=fnd&pg=PA122&dq=Jacob+Bernoulli+was+born+in+%5B%5BBasel%5D%5D+in+the+%5B%5BOld+Swiss+Confederacy&ots=f3r6a8MFSv&sig=Yoyxfg9FcSla_ItyvClK26A67Bw&redir_esc=y#v=onepage&q&f=false|editor-last1= Magill|editor-first1=Frank N.|title=The 17th and 18th Centuries: Dictionary of World Biography, Volume 4,|page=122|url=https://books.google.co.uk/books?id=HaHdAAAAQBAJ&dq=Jacob+Bernoulli+was+born+in+%5B%5BBasel%5D%5D+in+the+%5B%5BOld+Swiss+Confederacy&lr=&source=gbs_navlinks_s |publication-place=Oxon|publisher=Routledge|publication-date=13 September 2013|isbn=1135924147|via=[[Google Scholar]]}}</ref> the son and grandson of [[Protestant]]<ref>{{Cite journal|last1=Peiffer|first1=Jeanne|date=June 2006 |url=https://ftp.gwdg.de/pub/misc/EMIS/journals/JEHPS/Novembre2006/Peifferanglais3.pdf|title=Jacob Bernoulli, teacher and rival of his brother Johann|journal=Electronic Journal for History of Probability and Statistics|volume=2|issue=1|pages=3}}</ref> spice merchants on his fathers side, his mother was born into a family engaged in banking and city governing.<ref>{{cite book|last1=Suzuki|first1=J|date=2007|veditors=Hockey  T ''et al''|title=The Biographical Encyclopedia of Astronomers.|url=https://link.springer.com/referenceworkentry/10.1007/978-0-387-30400-7_141|publication-place=New York, NY.|publisher=Springer|doi=10.1007/978-0-387-30400-7_141|access-date=1 March 2025|isbn=978-0-387-30400-7|via=[[Qwant]]}}</ref>

Following his father's wish, he studied [[theology]] and entered the ministry. But contrary to the desires of his parents,<ref name =HLS>{{cite web |url=http://www.hls-dhs-dss.ch/textes/d/D23988.php |title=Bernoulli, Jacob |last=Nagel |first=Fritz |date=11 June 2004 |publisher=Historisches Lexikon der Schweiz |access-date=20 May 2016}}</ref> he also studied [[mathematics]] and [[astronomy]]. He traveled throughout [[Europe]] 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]], [[Robert Boyle]], and [[Robert Hooke]]. During this time he also produced an incorrect theory of [[comet]]s.
[[File:Acta Eruditorum - X astronomia, 1682 – BEIC 13349171.jpg|thumb|Image from ''[[Acta Eruditorum]]'' (1682) wherein was published the critique of Bernoulli's ''Conamen novi systematis cometarum'']]

Bernoulli returned to Switzerland, and began teaching mechanics at the [[University of Basel]] from 1683. His doctoral dissertation ''Solutionem tergemini problematis'' was submitted in 1684.<ref>{{cite book |last1=Kruit |first1=Pieter C. van der |title=Jan Hendrik Oort: Master of the Galactic System |date=2019 |publisher=Springer |isbn=978-3-030-17801-7 |page=639 |url=https://books.google.com/books?id=XwSjDwAAQBAJ&pg=PA639 |language=en}}</ref> It appeared in print in 1687.<ref>{{cite book |last1=Bernoulli |first1=Jakob |title=Die Werke von Jakob Bernoulli: Bd. 2: Elementarmathematik |date=2006 |publisher=Springer Science & Business Media |isbn=978-3-7643-1891-8 |page=92 |url=https://books.google.com/books?id=CYvAH41605QC&pg=PA92 |language=it}}</ref>

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]]'' were discovered.

People believe he was appointed professor of mathematics at the [[University of Basel]] in 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brother [[Johann Bernoulli]] 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]]" published in ''[[Acta Eruditorum]]''. They also studied the publications of [[Ehrenfried Walther von Tschirnhaus|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.<ref>{{cite web |url=http://www.jehps.net/Novembre2006/Peifferanglais3.pdf |title=Jacob Bernoulli |last= Pfeiffer |first=Jeanne |date=November 2006 |publisher=Journal Électronique d'Histoire des Probabilités et de la Statistique |access-date=20 May 2016}}</ref> By 1697, the relationship had completely broken down.

The lunar crater [[Bernoulli (crater)|Bernoulli]] 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]] 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 <math>\sum{\frac{1}{n}}</math> diverges, which Bernoulli believed were new but they had actually been proved by [[Pietro Mengoli]] 40 years earlier and was proved by Nicole Oresme in the 14th century already.<ref>D. J. Struik (1986) A Source Book In Mathematics, 1200-1800, p. 320</ref> Bernoulli could not find a closed form for <math>\sum{\frac{1}{n^2}}</math>, but he did show that it converged to a finite limit less than 2. [[Euler]] was the first to find [[Basel problem|the limit of this series]] in 1737. Bernoulli also studied [[#Discovery of the mathematical constant e|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 [[Tautochrone curve|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]]. Jacob Bernoulli's paper of 1690 is important for the history of calculus, since the term [[integral]] appears for the first time with its integration meaning. In 1696, Bernoulli solved the equation, now called the [[Bernoulli differential equation]],

:<math> y' = p(x)y + q(x)y^n. </math>

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]], the [[logarithmic spiral]] and [[epicycloids]] around 1692. The [[lemniscate of Bernoulli]] 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.

[[File:Bernoulli - Ars conjectandi, 1713 - 058.tif|thumb|''Ars conjectandi'', 1713 (Milano, [[Fondazione Mansutti]]).]]
Bernoulli's most original work was ''[[Ars Conjectandi]]'', 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 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 [[Probability theory|mathematical probability]], 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 {{mvar|[[e (mathematical constant)|e]]}} by studying a question about [[compound interest]] which required him to find the value of the following expression (which is in fact {{math|''e''}}):<ref>Jacob Bernoulli (1690) "Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685" (Some questions about interest, with a solution of a problem about games of chance, proposed in the ''Journal des Savants'' (''Ephemerides Eruditorum Gallicanæ''), in the year (anno) 1685.**), ''Acta eruditorum'', pp. 219–23.  [https://books.google.com/books?id=s4pw4GyHTRcC&pg=PA222 On p. 222], Bernoulli poses the question:  ''"Alterius naturæ hoc Problema est:  Quæritur, si creditor aliquis pecuniæ summam fænori exponat, ea lege, ut singulis momentis pars proportionalis usuræ annuæ sorti annumeretur; quantum ipsi finito anno debeatur?"'' (This is a problem of another kind:  The question is, if some lender were to invest [a] sum of money [at] interest, let it accumulate, so that [at] every moment [it] were to receive [a] proportional part of [its] annual interest; how much would he be owed [at the] end of [the] year?)  Bernoulli constructs a power series to calculate the answer, and then writes: ''" ... quæ nostra serie [mathematical expression for a geometric series] &c. major est.  ... si ''a''=''b'', debebitur plu quam {{sfrac|2|1|2}}''a''  & minus quam 3''a''."'' ( ... which our series [a geometric series] is larger [than]. ... if ''a''=''b'', [the lender] will be owed more than {{sfrac|2|1|2}}''a'' and less than 3''a''.)  If ''a''=''b'', the geometric series reduces to the series for ''a'' × ''e'', so 2.5 < ''e'' < 3.  (** The reference is to a problem which Jacob Bernoulli posed and which appears in the ''Journal des Sçavans'' of 1685 at the bottom of [http://gallica.bnf.fr/ark:/12148/bpt6k56536t/f307.image.langEN page 314.])</ref><ref>{{Cite web|url = http://www-history.mcs.st-and.ac.uk/HistTopics/e.html|title = The number e |author1=J J O'Connor |author2=E F Robertson |publisher = St Andrews University|access-date = 2 November 2016}}</ref>

:<math>\lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n</math>

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 $1.00×1.5<sup>2</sup>&nbsp;=&nbsp;$2.25. Compounding quarterly yields $1.00×1.25<sup>4</sup>&nbsp;=&nbsp;$2.4414..., and compounding monthly yields $1.00×(1.0833...)<sup>12</sup>&nbsp;=&nbsp;$2.613035....

Bernoulli noticed that this sequence approaches a limit (the [[Compound interest#Force of interest|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 {{math|''n''}} as the number of compounding intervals, with interest of 100% / {{math|''n''}} in each interval, the limit for large {{math|''n''}} is the number that [[Leonhard Euler|Euler]] later named {{math|''e''}}; with ''continuous'' compounding, the account value will reach $2.7182818.... More generally, an account that starts at $1, and yields (1+{{math|R}}) dollars at [[Interest#Compound interest|compound interest]], will yield {{math|''e''}}<sup>{{math|R}}</sup> dollars with continuous compounding.

==Tombstone==
[[Image:Basel - Grabstein Bernoulli.jpg|thumb|Jacob Bernoulli's tombstone in [[Basel Münster]]]]
Bernoulli wanted a [[logarithmic spiral]] and the motto ''[[Eadem mutata resurgo]]'' ('Although changed, I rise again the same') engraved on his tombstone. He wrote that the [[Self-similarity|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.<ref>{{cite book|last=Livio|first=Mario|author-link=Mario Livio|title=The Golden Ratio: The Story of Phi, the World's Most Astonishing Number|url=https://books.google.com/books?id=bUARfgWRH14C|orig-year=2002|edition=First trade paperback|year=2003|publisher=[[Random House|Broadway Books]]|location=New York City|isbn=0-7679-0816-3|pages=116–17}}</ref>

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 ==
* {{cite book|language=la|publisher=apud Henr. Wetstenium|title=Conamen novi systematis cometarum|location=Amstelaedami|year=1682|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=17384&custom_att_2=simple_viewer&search_terms=DTL4&pds_handle=}} (title roughly translates as "A new hypothesis for the system of comets".)
* {{cite book|language=la|publisher=apud Henricum Wetstenium|title=De gravitate aetheris|location=Amstelaedami|year=1683|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=1216514&custom_att_2=simple_viewer&search_terms=DTL5&pds_handle=}}
*''Ars conjectandi, opus posthumum'', Basileae, impensis Thurnisiorum Fratrum, 1713.
* {{Cite book|title=Opera|volume=1|publisher=héritiers Cramer & frères Philibert|location=Genève|year=1744|language=la|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=12199963}}
** {{Cite book|title=Opera|volume=2|publisher=héritiers Cramer & frères Philibert|location=Genève|year=1744|language=la|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=12202240}}

<gallery>
Bernoulli - De gravitate aetheris, 1683 - 1216514.jpg|''De gravitate aetheris'', 1683
Bernoulli, Jakob – Opera, vol 1, 1744 – BEIC 12199963.jpg|''Opera'', vol 1, 1744
</gallery>

==Notes==
{{notelist}}

==References==
{{reflist}}

==Further reading==
*{{DSB
|first=J.E.
|last=Hoffman
|title=Bernoulli, Jakob (Jacques) I
|volume=2
|pages=46–51
}}
*{{cite book |first=I. |last=Schneider |chapter=Jakob Bernoulli ''Ars conjectandi'' (1713) |chapter-url=https://books.google.com/books?id=UdGBy8iLpocC&pg=PA88 |editor1-first=Ivor |editor1-last=Grattan-Guinness |editor1-link=Ivor Grattan-Guinness |title=Landmark Writings in Western Mathematics 1640–1940 |url=https://books.google.com/books?id=UdGBy8iLpocC |year=2005 |publisher=Elsevier |isbn=978-0-08-045744-4 |pages=88–104 }}

== External links ==
*{{Commons category-inline}}
{{wikiquote}}
* {{MathGenealogy|id=54440|title=Jacob Bernoulli}}
* {{MacTutor Biography|id=Bernoulli_Jacob|title=Jacob Bernoulli}}
* {{cite web |first=Jacobi |last=Bernoulli |title=Tractatus de Seriebus Infinitis |url=http://www.kubkou.se/pdf/mh/jacobB.pdf }}
* {{ScienceWorldBiography | urlname=BernoulliJakob | title=Bernoulli, Jakob (1654–1705)}}
* Gottfried Leibniz and Jakob Bernoulli [http://cerebro.xu.edu/math/Sources/JakobBernoulli/jakob%20and%20leibniz.pdf Correspondence Regarding the Art of Conjecturing"] {{Webarchive|url=https://web.archive.org/web/20160406004155/http://cerebro.xu.edu/math/Sources/JakobBernoulli/jakob%20and%20leibniz.pdf |date=2016-04-06 }}

{{Bernoulli family}}
{{Authority control}}

{{DEFAULTSORT:Bernoulli, Jakob}}
[[Category:1655 births]]
[[Category:1705 deaths]]
[[Category:17th-century apocalypticists]]
[[Category:17th-century Swiss mathematicians]]
[[Category:18th-century apocalypticists]]
[[Category:18th-century writers in Latin]]
[[Category:18th-century male writers]]
[[Category:18th-century Swiss mathematicians]]
[[Category:Bernoulli family|Jacob]]
[[Category:Burials at Basel Münster]]
[[Category:Members of the French Academy of Sciences]]
[[Category:Number theorists]]
[[Category:Scientists from Basel-Stadt]]
[[Category:Probability theorists]]
[[Category:Swiss mathematicians]]