Solutionem tergemini problematis arithmetici, geometrici et astronomici... (Solution to a triple problem in arithmetics, geometry and astronomy...)(1684)
Jacob Bernoulli[a] (also known asJames in English orJacques in French; 6 January 1655 [O.S. 27 December 1654] – 16 August 1705) was a Swiss mathematician. He sided withGottfried Wilhelm Leibniz during theLeibniz–Newton calculus controversy and was an early proponent of Leibniziancalculus, to which he made numerous contributions. A member of theBernoulli family, he, along with his brotherJohann, 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 numbers in his workArs Conjectandi.[3]
Jacob Bernoulli was born inBasel in theSwiss Confederation;[4] the son and grandson ofProtestant[5] spice merchants on his father's side, his mother was born into a family engaged in banking and city governing.[6]
Following his father's wish, he studiedtheology and entered the ministry. But contrary to the desires of his parents,[7] he also studiedmathematics andastronomy. He traveled throughoutEurope from 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 Basel from 1683. His doctoral dissertationSolutionem tergemini problematis was submitted in 1684.[8] It appeared in print in 1687.[9]
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 Géométrie andFrans van Schooten's supplements of it. He also studiedIsaac Barrow andJohn 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 Conjectandi were discovered.
People believe he was appointed professor of mathematics at theUniversity of Basel in 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brotherJohann 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 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.[10] By 1697, the relationship had completely broken down.
The lunar craterBernoulli is also named after him jointly with his brother Johann.
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 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 diverges, which Bernoulli believed were new but they had actually been proved byPietro Mengoli 40 years earlier and was proved by Nicole Oresme in the 14th century already.[11] Bernoulli could not find a closed form for, but he did show that it converged to a finite limit less than 2.Euler was the first to findthe limit of this series in 1737. Bernoulli also studiedthe exponential series which came out of examining compound interest.
In May 1690, in a paper published inActa Eruditorum, Jacob Bernoulli showed that the problem of determining theisochrone 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 callseparation of variables. Jacob Bernoulli's paper of 1690 is important for the history of calculus, since the termintegral appears for the first time with its integration meaning. In 1696, Bernoulli solved the equation, now called theBernoulli differential equation,
Jacob Bernoulli also discovered a general method to determineevolutes 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 theparabola, thelogarithmic spiral andepicycloids around 1692. Thelemniscate 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.
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 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 termBernoulli trial resulted 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 and 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.
In 1683, Bernoulli discovered the constante by studying a question aboutcompound interest which required him to find the value of the following expression (which is in facte):[12][13]
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. Usingn as the number of compounding intervals, with interest of 100% /n in each interval, the limit for largen is the number thatEuler later namede; withcontinuous compounding, the account value will reach $2.7182818.... More generally, an account that starts at $1, and yields (1+R) dollars atcompound interest, will yieldeR dollars with continuous compounding.
Bernoulli wanted alogarithmic spiral and the mottoEadem mutata resurgo ('Although changed, I rise again the same') engraved on his tombstone. He wrote that theself-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 anArchimedean spiral was engraved rather than a logarithmic one.[14]
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.
Conamen novi systematis cometarum (in Latin). Amstelaedami: apud Henr. Wetstenium. 1682. (title roughly translates as "A new hypothesis for the system of comets".)
^Pfeiffer, Jeanne (November 2006)."Jacob Bernoulli"(PDF). Journal Électronique d'Histoire des Probabilités et de la Statistique. Retrieved20 May 2016.
^D. J. Struik (1986) A Source Book In Mathematics, 1200-1800, p. 320
^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 theJournal des Savants (Ephemerides Eruditorum Gallicanæ), in the year (anno) 1685.**),Acta eruditorum, pp. 219–23.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. ... sia=b, debebitur plu quam2+1/2a & minus quam 3a." ( ... which our series [a geometric series] is larger [than]. ... ifa=b, [the lender] will be owed more than2+1/2a and less than 3a.) Ifa=b, the geometric series reduces to the series fora ×e, so 2.5 <e < 3. (** The reference is to a problem which Jacob Bernoulli posed and which appears in theJournal des Sçavans of 1685 at the bottom ofpage 314.)
^J J O'Connor; E F Robertson."The number e". St Andrews University. Retrieved2 November 2016.
