Jacob Bernoulli

For other family members namedJacob,seeBernoulli family.

Jacob Bernoulli
Born	(1655-01-06)6 January 1655 Basel,Switzerland
Died	16 August 1705(1705-08-16)(aged 50) Basel, Switzerland
Education	University of Basel (D.Th., 1676; Dr. phil. hab., 1684)
Known for	Bernoulli differential equation Bernoulli numbers Bernoulli's formula Bernoulli polynomials Bernoulli map Bernoulli trial Bernoulli process Bernoulli scheme Bernoulli operator Hidden Bernoulli model Bernoulli sampling Bernoulli distribution Bernoulli random variable Bernoulli's Golden Theorem Bernoulli's inequality Lemniscate of Bernoulli
Scientific career
Fields	Mathematics,mechanics
Institutions	University of Basel
Theses	Primi et Secundi Adami Collatio(1676) Solutionem tergemini problematis arithmetici, geometrici et astronomici... (Solution to a triple problem in arithmetics, geometry and astronomy...)(1684)
Doctoral advisor	Peter Werenfels (1676 thesis advisor)
Other academic advisors	Gottfried Wilhelm Leibniz(epistolary correspondent)
Doctoral students	Jacob Hermann Nicolaus I Bernoulli
Other notable students	Johann Bernoulli
Notes
Brother ofJohann Bernoulli

Jacob Bernoulli(also known asJamesin English orJacquesin French; 6 January 1655 [O.S.27 December 1654] – 16 August 1705) was one of the many prominentmathematiciansin the SwissBernoulli family.He sided withGottfried Wilhelm Leibnizduring theLeibniz–Newton calculus controversyand was an early proponent of Leibniziancalculus,which he made numerous contributions to; along with his brotherJohann,he was one of the founders of thecalculus of variations.He also discovered the fundamental mathematical constante.However, his most important contribution was in the field ofprobability,where he derived the first version of thelaw of large numbersin his workArs Conjectandi.

Biography

Jacob Bernoulli was born inBaselin theOld Swiss Confederacy.Following his father's wish, he studiedtheologyand entered the ministry. But contrary to the desires of his parents,he also studiedmathematicsandastronomy.He traveled throughoutEuropefrom 1676 to 1682, learning about the latest discoveries in mathematics and the sciences under leading figures of the time. This included the work ofJohannes Hudde,Robert Boyle,andRobert Hooke.During this time he also produced an incorrect theory ofcomets.

Image fromActa Eruditorum(1682) wherein was published the critique of Bernoulli'sConamen novi systematis cometarum

Bernoulli returned to Switzerland, and began teaching mechanics at theUniversity of Baselfrom 1683. His doctoral dissertationSolutionem tergemini problematiswas 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, includingChristiaan Huygens'sDe ratiociniis in aleae ludo,Descartes'La GéométrieandFrans van Schooten's supplements of it. He also studiedIsaac BarrowandJohn 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 upArs Conjectandiwere discovered.

He was appointed professor of mathematics at theUniversity of Baselin 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brotherJohann Bernoullion 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 inActa Eruditorum.They also studied the publications ofvon 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 craterBernoulliis 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 oninfinite seriesand 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$\sum {\frac {1}{n}}$diverges, which Bernoulli believed were new but they had actually been proved byPietro Mengoli40 years earlier and was proved by Nicole Oresme in the 14th century already.Bernoulli could not find a closed form for$\sum {\frac {1}{n^{2}}}$,but he did show that it converged to a finite limit less than 2.Eulerwas the first to findthe limit of this seriesin 1737. Bernoulli also studiedthe exponential serieswhich came out of examining compound interest.

In May 1690, in a paper published inActa Eruditorum,Jacob Bernoulli showed that the problem of determining theisochroneis 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 callseparation of variables.Jacob Bernoulli's paper of 1690 is important for the history of calculus, since the termintegralappears for the first time with its integration meaning. In 1696, Bernoulli solved the equation, now called theBernoulli differential equation,

$y'=p(x)y+q(x)y^{n}.$

Jacob Bernoulli also discovered a general method to determineevolutesof a curve as the envelope of its circles of curvature. He also investigated caustic curves and in particular he studied these associated curves of theparabola,thelogarithmic spiralandepicycloidsaround 1692. Thelemniscate of Bernoulliwas 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.

Ars conjectandi,1713 (Milano,Fondazione Mansutti).

Bernoulli's most original work wasArs 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 ofcombinatorics,in particular the work of van Schooten, Leibniz, and Prestet, as well as the use ofBernoulli numbersin 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 termBernoulli trialresulted from this work.

In the last part of the book, Bernoulli sketches many areas ofmathematical 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 constante

In 1683, Bernoulli discovered the constanteby studying a question aboutcompound interestwhich required him to find the value of the following expression (which is in facte):

$\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$

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 (theforce of interest) for more and smaller compounding intervals. Compounding weekly yields $2.692597..., while compounding daily yields $2.714567..., just two cents more. Usingnas the number of compounding intervals, with interest of 100% /nin each interval, the limit for largenis the number thatEulerlater namede;withcontinuouscompounding, the account value will reach $2.7182818.... More generally, an account that starts at $1, and yields (1+R) dollars atcompound interest,will yielde^Rdollars with continuous compounding.

Tombstone

Jacob Bernoulli's tombstone inBasel Münster

Bernoulli wanted alogarithmic spiraland the mottoEadem mutata resurgo('Although changed, I rise again the same') engraved on his tombstone. He wrote that theself-similarspiral "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 anArchimedean spiralwas 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

• Conamen novi systematis cometarum(in Latin). Amstelaedami: apud Henr. Wetstenium. 1682.(title roughly translates as "A new hypothesis for the system of comets".)
• De gravitate aetheris(in Latin). Amstelaedami: apud Henricum Wetstenium. 1683.
• Ars conjectandi, opus posthumum,Basileae, impensis Thurnisiorum Fratrum, 1713.
• Opera(in Latin). Vol. 1. Genève: héritiers Cramer & frères Philibert. 1744.
  • Opera(in Latin). Vol. 2. Genève: héritiers Cramer & frères Philibert. 1744.

• 
  De gravitate aetheris,1683
• 
  Opera,vol 1, 1744

Further reading

• Hoffman, J.E. (1970–1980). "Bernoulli, Jakob (Jacques) I".Dictionary of Scientific Biography.Vol. 2. New York: Charles Scribner's Sons. pp. 46–51.ISBN978-0-684-10114-9.
• Schneider, I. (2005)."Jakob BernoulliArs conjectandi(1713) ".InGrattan-Guinness, Ivor(ed.).Landmark Writings in Western Mathematics 1640–1940.Elsevier. pp. 88–104.ISBN978-0-08-045744-4.

External links

• Media related toJakob Bernoulliat Wikimedia Commons

• Jacob Bernoulliat theMathematics Genealogy Project
• O'Connor, John J.;Robertson, Edmund F.,"Jacob Bernoulli",MacTutor History of Mathematics Archive,University of St Andrews
• Bernoulli, Jacobi."Tractatus de Seriebus Infinitis"(PDF).
• Weisstein, Eric Wolfgang(ed.)."Bernoulli, Jakob (1654–1705)".ScienceWorld.
• Gottfried Leibniz and Jakob BernoulliCorrespondence Regarding the Art of Conjecturing "Archived2016-04-06 at theWayback Machine