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| :''For other family members named Jacob, see [[Bernoulli family]].''
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| {{refimprove|date=June 2010}}
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| {{Infobox scientist
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| | this page has been verified by kaleb batman
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| name = Jakob Bernoulli
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| | image = Jakob_Bernoulli.jpg|200px
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| | image_size = 200px
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| | caption = Jakob Bernoulli
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| | birth_date = {{birth date|df=yes|1655|1|6}}
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| | birth_place = [[Basel]], [[Old Swiss Confederacy|Switzerland]]
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| | death_date = {{death date and age|df=yes|1705|8|16|1655|1|6}}
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| | death_place = Basel, [[Old Swiss Confederacy|Switzerland]]
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| | residence = [[Switzerland]]
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| | field = [[Mathematician]]
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| | work_institution = [[University of Basel]]
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| | alma_mater = [[University of Basel]]
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| | doctoral_advisor = [[Nicolas Malebranche]]<br>[[Peter Werenfels]]
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| | doctoral_students = [[Johann Bernoulli]]<br>[[Jacob Hermann (mathematician)|Jacob Hermann]]<br>[[Nicolaus I Bernoulli]]
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| | influences = [[Gottfried Leibniz]]
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| | 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]]
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| | religion = [[Calvinist]]
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| | footnotes= Brother of [[Johann Bernoulli]]
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| }}
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| '''Jacob Bernoulli''' (also known as '''James''' or '''Jacques'''; {{OldStyleDateDY|6 January|1655|27 December 1654}} – 16 August 1705) was one of the many prominent [[mathematicians]] in the [[Bernoulli family]]. He was an early proponent of Leibnizian calculus and had sided with Leibniz during the [[Leibniz–Newton calculus controversy]]. He is known for his numerous contributions to [[calculus]], and along with his brother [[Johann Bernoulli|Johann]], was one of the founders the [[calculus of variations]]. 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>
| | This standalone DVR encourage any standard SATA hard disk drive, or you'll be able to have us install a hard drive to suit your needs, to ensure it is ready to go if it ships. The other item that I found while reviewing the item, that the program didn't offer is really a per user charge of blocked best [http://www.Bestpampers.com/members/duberberube/activity/44096/ affordable security] camera system web-site categories.<br><br>An individual that is writing a novel also needs to come up with the primary conflict. Prioritise for the approach that could possibly use all the data you've got chosen. again - [http://www.hellotw.com/gate/big5/cctvdvrreviews.com www.hellotw.com] - cctv dvr features You might want to create about your techniques and turn them in a series of articles to the newcomers which are starting out with this business. For the reason that the topic is to broad, it would definitely need a large amount of pages 16 channel dvr cameras to ensure that you to dvr cctv software for mac publish about in thoroughly. |
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| ==Biography==
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| Jacob Bernoulli was born in [[Basel]], [[Switzerland]]. Following his father's wish, he studied [[theology]] and entered the ministry. But contrary to the desires of his parents, 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 [[Hudde]], [[Robert Boyle]], and [[Robert Hooke]]. During this time he also produced an incorrect theory of comets.
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| Bernoulli returned to Switzerland and began teaching mechanics at the University in Basel from 1683. In 1684 he married Judith Stupanus; and 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]]' ''Geometrie'' 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 much of the results that was to make up ''[[Ars Conjectandi]]'' was discovered.
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| 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, itemque Tangentibus...'' 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 the first to try to understand and apply Leibniz's theories.
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| 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.
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| Jacob Bernoulli died in 1705. Bernoulli chose a figure of a [[logarithmic spiral]] and the motto ''[[Eadem mutata resurgo]]'' ''("Changed and yet the same, I rise again")'' for his gravestone; the spiral executed by the stonemasons was, however, an [[Archimedean spiral]], “[Jacques Bernoulli] wrote that the logarithmic 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’.” (Livio 2002: 116). His grave is in [[Basel Münster|Basel Munster or Cathedral]] where the gravestone shown below is located.
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| The lunar crater [[Bernoulli (crater)|Bernoulli]] is also named after him jointly with his brother Johann.
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| ==Important works==
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| [[Image:Basler Muenster Bernoulli.jpg|thumb|left|Jacob Bernoulli's grave.]]
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| 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.
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| 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 fundamental result that <math>\sum 1/n</math> diverges, which Bernoulli believed were new but they had actually been proved by Mengoli 40 years earlier. Bernoulli could not find a closed form for <math>\sum 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 sum of this series]] in 1737. Bernoulli also studied [[Jacob_Bernoulli#Discovery_of_the_mathematical_constant_e|the exponential series]] which came out of examining compound interest.
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| 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]]. 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]],
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| :<math> y' = p(x)y + q(x)y^n. </math>
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| 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.
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| Jacob 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. In the book Bernoulli reviewed work of others on probability, in particular work by van Schooten, Leibniz, and Prestet. The [[Bernoulli numbers]] appear in the book in a discussion of the exponential series. Many examples are given on how much one would expect to win playing various game of chance. The term [[Bernoulli trial]] resulted from this work. There are interesting thoughts on what probability really is:
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| <blockquote>
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| ... 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; law of large numbers ...
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| </blockquote>
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| 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.
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| ==Discovery of the mathematical constant e==
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| Bernoulli discovered the constant [[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''}}):
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| :<math>\lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n</math>
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| 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<sup>4</sup> = $2.4414..., and compounding monthly yields $1.00×(1.0833...)<sup>12</sup> = $2.613035....
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| 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 came to be known as {{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#Simple interest|simple interest]], will yield {{math|''e''}}<sup>{{math|R}}</sup> dollars with continuous compounding.
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| ==Translation of Latin inscription on Bernouilli's tomb==
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| :IACOBUS BERNOULLI
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| :MATHEMATICUS INCOMPARABILIS
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| :ACAD. BASIL.
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| :VLTRA XVIII ANNOS PROF.
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| :ACADEM. ITEM REGIAE PARIS. ET BEROLIN.
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| :SOCIUS
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| :EDITIS LUCUBRAT. INLUSTRIS.
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| :MORBO CHRONICO
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| :MENTE AD EXTREMUM INTEGRA
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| :ANNO SALUT. MDCCV. D. XVI. AUGUSTI
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| :AETATIS L. M. VII
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| :EXTINCTUS
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| :RESURRECT. PIOR. HIC PRAESTOLATUR
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| :IUDITHA STUPANA
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| :XX ANNOR. UXOR
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| :CUM DUOBUS LIBERIS
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| :MARITO ET PARENTI
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| :EHEU DESIDERATISS.
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| :H.M.P.
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| :James Bernoulli, the incomparable mathematician.
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| :Professor at the University of Basel For more than 18 years;
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| :member of the Royal Academies of Paris and Berlin; famous for his writings.
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| :Of a chronic illness, of sound mind to the end;
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| :succumbed in the year of grace 1705, the 16th of August, at the age of 50 years and 7 months, awaiting the resurrection.
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| :Judith Stupanus,
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| :his wife for 20 years,
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| :and his two children have erected a monument to the husband and father they miss so much.
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| *{{DSB
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| |first=J.E.
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| |last=Hoffman
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| |title=Bernoulli, Jakob (Jacques) I
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| |volume=2
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| |pages=46–51
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| }}
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| *{{cite book |first=Schneider |last=I. |chapter=Jakob Bernoulli ''Ars conjectandi'' (1713) |chapterurl=http://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=http://books.google.com/books?id=UdGBy8iLpocC |year=2005 |publisher=Elsevier |isbn=978-0-08-045744-4 |pages=88–104 }}
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| *{{cite book |first=Mario |last=Livio |title=The golden ratio: the story of phi, the world's most astonishing number |url=http://books.google.com/books?id=9kvvAAAAMAAJ |year=2002 |publisher=Broadway Books |isbn=978-0-7679-0815-3}}
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| == External links ==
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| * {{MathGenealogy|id=54440}}
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| * {{MacTutor Biography|id=Bernoulli_Jacob}}
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| * {{cite web |first=Jakob |last=Bernoulli |title=Tractatus de Seriebus Infinitis |url=http://www.kubkou.se/pdf/mh/jacobB.pdf |format=PDF}}
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| * {{ScienceWorldBiography | urlname=BernoulliJakob | title=Bernoulli, Jakob (1654–1705)}}
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| {{Bernoulli family}}
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| {{Authority control |VIAF=14793931 |LCCN=n/83/196756 }}
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| {{Persondata <!-- Metadata: see [[Wiondata]] -->
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| |NAME= Bernoulli, Jacob
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| |ALTERNATIVE NAMES=
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| |SHORT DESCRIPTION= Swiss cleric and [[mathematician]]
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| |DATE OF BIRTH= 6 January 1655
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| |PLACE OF BIRTH= [[Basel]], Switzerland
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| |DATE OF DEATH= 16 August 1705
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| |PLACE OF DEATH= Basel, Switzerland
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| }}
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| {{DEFAULTSORT:Bernoulli, Jakob}}
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| [[Category:1655 births]]
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| [[Category:1705 deaths]]
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| [[Category:17th-century mathematicians]]
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| [[Category:17th-century Swiss people]]
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| [[Category:18th-century Latin-language writers]]
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| [[Category:18th-century Swiss mathematicians]]
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| [[Category:Number theorists]]
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| [[Category:People from Basel-Stadt]]
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| [[Category:Probability theorists]]
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| [[Category:Swiss Calvinist and Reformed ministers]]
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| [[Category:Swiss mathematicians]]
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| [[Category:Swiss scientists]]
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| [[Category:Apocalypticists]]
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| [[Category:Burials at Basel Münster]]
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| [[Category:Bernoulli family]]
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