THE BERNOULLI FAMILY

    One of the most distinguished families in the history of mathematics and science is the Bernoulli family of Switzerland, which from the late secenteenth century on, producced an unusual number of capable mathematicians and scientists.  The family record starts with the two brothers, Jakob Bernoulli(1654-1705) and Johann Bernoulli (1667-1748), some of whose mathematical accomplishments have already been mentioned in this

book.  These two men gave up earlier vocational interests and became mathematicians when Leibniz' papers began to appear in the Acta eruditorum.     They were among the first mathematicians to realize the surprising power of the calculcus and to apply the tool to a great diversity of problems.  From 1687 until his death, Jakob occupied the mathematics chair at Basel University.  Johann, in 1697, became a professor at Groningen University, and then, on Jakob's death in 1705, succeeded his brother in the chair at Basel University, to remain there for the rest of his life.   The two brothers, often bitter rivals, maintained an almost constant exchange of ideas with Leibniz and with each other.
     One of the first mathematicians to work in the calculus of variations.   He was also one of the early students of mathematical probability; his book in this field, the Ars conjectandi, was posthumously published in 1713. Several things in mathematics now bear Jakob Bernoulli's name. Among these are the Bernoulli distribution and Bernolli theorem of statistics and probability theory; the Bernoulli equation, met by every student of a first course in differential equations;the Bernoulli numbers, and Bernoulli polynomials first course in the calculus. In jakob Bernoulli's soulution to the problem of the isochrone curve, which was published in the Acta eruditorum, in 1690. we meet for the first time the word integral in a caculus sense. Leibniz had called the integral calculus calculus summatorius; in 1696, Leibniz and Johann Bernoulli agreed to call it calculus calculus integralis. Jakob Bernoulli was struck by the way the equiangular spiral reproduces itselt under a bariety of thransformations and asked, in imitation of Archimedes, that such a spiral be engraved on his tombstone, along with the inscription "Eadem mutata resurgo" ("Though changed, I arise again the same.")

    Johann Bernoulli was an even more prolific contributor to mathematics than was his brother Jakob, Though he was a jealous and cantankerous man, he was one of the most successful teachers of his time. He greatly enriched the cajculus and was very influential in making the power of the new subject apprecianted in contincntal Europe. As we have seen, if was his material that the Marquis de I;Hospital (1661-1704), under a curious

financial agreement with Johann, assembled in 1696 into the first calculus textbook. In this way, the familiar metgod of evaluationg the indeterminate form 0/0 became incorrectly known in later calculus texts as I'Hospital's rule.     Johann Bernoulli had three sons, Nicolaus (1695-1726), Daniel(1700- 1782), and Johann II (1710-1790), all of whom won renown as eighteenthcentury mathematicians and scientists. Nicolaus, who showed great promise in the field of mathematics, was called to the St.Petersburg Academy, where he unfortunately died by drowining, only eight months later. He wrote on curves, differential equations, and probability. A problem in probability, which he proposed from St. Peresburg, later became known as the Petersburg paradox. The problem is:If A receives a penny when a head appears on the first toss of a coin, two pennies if a head does not appear until second toss, four pennies if a head does not appear until the third toss, and so on, what is A's expectation? Mathematical theory shows that A's expectation is infinite, which seems a paradoxical result. The problem was incestigated by Nicolaus' brother Daniel, who succeeded Nicolaus at St. Petersburg. Daniel returned to Basel seven years later. He was the most famous of Johann's three sons, and devoted most of his energies to probability, astronmy, physics and hydrodynamics. In probability he debised the concept of moral expectation, and in his Hydrodynamica, of 1738, appears the principle of hydrodynamics that bears his name in all present-day elementary physics texts.
    Johann II, the youngest of the three sons, studied law but spent his later years as a professor of mathematics at the University of Basel. He was particularly interested in the mathematical theory of heat and light.
    There was another eighteenth-century Nicolaus Bernoulli (1687-1759), a nephew of Jakob and Johann, who achieved some fame in mathematics. This Nicoiaus held, for a time, the chair of mathematics at Padua once filled by Galileo. He wrote extensively on geometry and differential equations. Later in life, he taught logic and law.
    Johann Bernoulli II had a son Johann III (1744-1807) who, like his father, studied law but then turned to mathematics. When barely nineteen years old, he was called as a professor of mathematics to the Berlin Academy. He wrote on astonomy, the doctrine of chance, recurring decimals, and indeterminate equations.
    LesserBernoulli descendants are Daniel II (1751-1834) and Jakob II (1759-1789), two other sons of Johann II, Christoph (1782-1834), a son of Daniel II, and Johann Gustav (1811-1863), a son of Christoph.