Jakob Bernoulli | Biography

Jakob Bernoulli, also known as Jacques, James, or Jakob I--to avoid any confusion with Jakob Bernoulli II--is one of the great names of 17th-century mathematics as well as the first member of the prodigiously mathematical Bernoulli family to attain international fame. Originally from the Spanish Netherlands, the Bernoullis moved to Basel, Switzerland, in 1583, to escape Spanish oppression; the first prominent Bernoulli was Nikolaus I, Jakob's father. A contemporary of the great German philosopher and mathematician Gottfried Wilhelm von Leibniz, with whom he maintained a correspondence, Bernoulli is known for his extraordinary contributions to calculus and the theory of probability.

Bernoulli earned a degree in theology in 1676, having studied mathematics and astronomy against his father's wishes. Employed as a tutor in Geneva in 1676, he later spent two years in France, where he studied the works of René Descartes. In 1681, Bernoulli traveled to the Netherlands and England, where he met Robert Boyle. During this period, Bernoulli wrote on a variety on scientific topics, including comets and gravity. Following his return to Basel 1682, Bernoulli founded a school for science and mathematics, where he lectured and conducted experiments in the field of mechanics. He also wrote articles for the two preeminent European scientific journals, Journal des sçavans and Acta eruditorum. In 1687, Bernoulli was named professor of mathematics at the University of Basel, a position he held until his death. Owing to his passion for learning, Bernoulli carefully studied the works of predecessors, such as Descartes, and contemporaries, such as Leibniz.

Leibniz's first discussion of the differential calculus, a six-page article, appeared in Acta eruditorum in 1684 (the presentation of the integral calculus was published two years later). While the term calculus refers to the process of calculation in general, Leibniz's calculus dealt with infinitesimals, quantities smaller than any definable finite quantity but larger than zero, and was therefore called infinitesimal calculus. It should be pointed out, however, that the term infinitesimal is no longer used in mathematical terminology, infinitesimal quantities being instead named limit values. Therefore, when mathematicians refer to calculus, the predicate infinitesimal is implied.

Calculus encompasses four distinct types. Differential calculus calculates derivatives.Integral calculus is the reverse of differential calculus--in other words, we use this type of calculus to determine a function when its derivative is known. Calculus of variation is used to find a function for which a given integral assumes a maximal or a minimal value. Differential equations are equations containing derivatives. Misunderstood by most of his colleagues, Leibniz's discovery nevertheless attracted a small following of mathematicians who realized the tremendous analytical power of calculus. Among Leibniz's followers, Bernoulli was among the first who completely grasped the essence of calculus, and he proceeded, in numerous contributions to Acta eruditorum, to develop the foundations of calculus.

In 1689, Bernoulli formulated the famous "Bernoulli inequality"--(1 + x)ⁿ 1 + nx, where x is real, x -1, and x 0, and n 1--which had in fact already been presented in 1670 by Isaac Barrow in his Lectiones geometriae. Bernoulli also provided solutions for several famous mathematical problems. For example, he solved the catenary equation. When a flexible, non-elastic cable is suspended from two fixed points, the shape it assumes as a result of gravity is a catenary. Bernoulli formulated an equation to refute the traditional hypothesis among mathematicians that the catenary is a parabola. In his investigations of the isochrone, a plane curve enabling an object to fall with uniform velocity, Bernoulli found an equation which demonstrated that the necessary curve is a semicubical parabola. In fact, while working on the isochrone problem, Bernoulli used the term integral, in an 1690 article, to denote the inverse of the differential calculus. Leibniz later agreed that calculus integralis was a more concise term than the original calculus summatorius .

Bernoulli was among several mathematicians, including his younger brother Johann Bernoulli and Leibniz, who worked on the brachistochrone (the term was derived from the Greek words brachystos, meaning the shortest and chronos, meaning time) problem. Essentially, the problem challenged mathematicians to find a curve of quickest descent between two given points A and B, assuming that Bdoes not lie right beneath A. While his brother correctly assumed that the required curve is a cycloid but offered an incorrect proof, Bernoulli provided the correct proof. The competition between the two brothers became so intense that Johann appropriated the brachistochrone solutionas his own, which was just one episode in their bitter struggle for preeminence in mathematics.

Much of Bernoulli's work was devoted to the study of curves. The lemniscate, or figure eight curve (from the Latin term lemniscus, meaning ribbon), was named after him. However, he was totally fascinated by the logarithmic spiral, which in nature can be seen in a cross section of the shell of a chambered nautilus. Bernoulli noticed that the logarithmic spiral has several unique properties, including self-similarity, which means that any portion, if scaled up or down, is congruent to other parts of the curve. In fact, Bernoulli was so taken by this spiral, also called spira mirabilis, or wonderful spiral, that he requested that it be engraved on his tombstone, along with the Latin inscription Eadem mutata resurgo ("Though changed, I arise again the same").

Bernoulli's research on probabilityis documented in his treatise Ars conjectandi ("The Art of Conjecture"), published posthumously in 1713. Building on earlier writings on the subject, including Girolamo Cardano's Liber de ludo aleae ("On Casting the Die"), published in 1663, the correspondence between Pierre de Fermat and Blaise Pascal, and Christiaan Huygens' 1656 book De ratiociniis in ludo aleae ("On Reasoning in Games of Chance"), Bernoulli created a work which scholars consider the substantial book on probability. In his book, to which he attached a treatise on infinite series (Tractatus de seriebus infinitis), Bernoulli presented his famous theorem which Siméon-Denis Poisson later named the "Law of Large Numbers." This law states that if a very large number of independent trials are made, then the observed proportion of successes for an event will, with probability close to 1, be very close to the theoretical probability of success for that event on each individual trial. Unfortunately, Bernoulli's treatise ends with his theorem, which he had hoped to use as the foundation of his project to apply the calculus of probability to a variety of fields, including demographics, politics, and economics.