Доклад: Jacob (Jacques) Bernoulli
Jacob (Jacques) Bernoulli
Born: 27 Dec 1654 in Basel, Switzerland
Died: 16 Aug 1705 in Basel, Switzerland
Jacob Bernoulli's father, Nicolaus Bernoulli
(16231708) inherited the spice business in Basel that had been set up by his
own father, first in Amsterdam and then in Basel. The family, of Belgium
origin, were refugees fleeing from persecution by the Spanish rulers of the
Netherlands. Philip, the King of Spain, had sent the Duke of Alba to the
Netherlands in 1567 with a large army to punish those opposed to Spanish rule,
to enforce adherence to Roman Catholicism, and to reestablish Philip's
authority. Alba set up the Council of Troubles which was a court that condemned
over 12000 people but most, like the Bernoulli family who were of the
Protestant faith, fled the country.
Nicolaus Bernoulli was an important citizen of Basel,
being a member of the town council and a magistrate. Jacob Bernoulli's mother
also came from an important Basel family of bankers and local councillors.
Jacob Bernoulli was the brother of
Johann Bernoulli and the uncle of
Daniel Bernoulli. He was compelled to study philosophy and theology by
his parents, which he greatly resented, and he graduated from the University of
Basel with a master's degree in philosophy in 1671 and a licentiate in theology
in 1676.
During the time that Jacob Bernoulli was taking his
university degrees he was studying mathematics and astronomy against the wishes
of his parents. It is worth remarking that this was a typical pattern for many
of the Bernoulli family who made a study of mathematics despite pressure to
make a career in other areas. However Jacob Bernoulli was the first to go down
this road so for him it was rather different in that there was no tradition of
mathematics in the family before Jacob Bernoulli. Later members of the family
must have been much influenced by the tradition of studying mathematics and
mathematical physics.
In 1676, after taking his theology degree, Bernoulli
moved to Geneva where he worked as a tutor. He then travelled to France
spending two years studying with the followers of Descartes who were led at this time by Malebranche. In 1681 Bernoulli travelled to the Netherlands where
he met many mathematicians including
Hudde. Continuing his studies with the leading mathematicians and
scientists of Europe he went to England where, among others, he met Boyle and
Hooke. At this time he was deeply interested in astronomy and produced a
work giving an incorrect theory of comets. As a result of his travels,
Bernoulli began a correspondence with many mathematicians which he carried on
over many years.
Jacob Bernoulli returned to Switzerland and taught
mechanics at the University in Basel from 1683, giving a series of important
lectures on the mechanics of solids and liquids. Since his degree was in
theology it would have been natural for him to turn to the Church, but although
he was offered an appointment in the Church he turned it down. Bernoulli's real
love was for mathematics and theoretical physics and it was in these topics
that he taught and researched. During this period he studied the leading
mathematical works of his time including Descartes's Géométrie
and van Schooten's additional material
in the Latin edition. Jacob Bernoulli also studied the work of Wallis and
Barrow and through these he became interested in infinitesimal geometry. Jacob began
publishing in the journal Acta Eruditorum which was established in Leipzig in
1682.
In 1684 Jacob Bernoulli married Judith Stupanus. They
were to have two children, a son who was given his grandfather's name of
Nicolaus and a daughter. These children, unlike many members of the Bernoulli
family, did not go on to become mathematicians or physicists.
One of the most significant events concerning the
mathematical studies of Jacob Bernoulli occurred when his younger brother, Johann Bernoulli, began to work on
mathematical topics. Johann Bernoulli
was told by his father to study medicine but while he was studying that topic
he asked his brother Jacob Bernoulli to teach him mathematics. Jacob Bernoulli
was appointed professor of mathematics in Basel in 1687 and 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 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.
Although Jacob and
Johann Bernoulli both worked on similar problems their relationship was
soon to change from one of collaborators to one of rivals. Johann Bernoulli's boasts were the first
cause of Jacob's attacks on him and Jacob wrote that Johann was his pupil whose
only achievements were to repeat what his teacher had taught him. Of course
this was a grossly unfair statement. Jacob Bernoulli continued to attack his
brother in print in a disgraceful and unnecessary fashion, particularly after
1697. However he did not reserve public criticism for his brother. He was
critical of the university authorities at Basel and again he was very public in
making critical statements that, as one would expect, left him in a difficult
situation at the university. Jacob probably felt that Johann was the more
powerful mathematician of the two and, this hurt since Jacob's nature meant
that he always had to feel that he was winning praise from all sides. Hofmann
writes in:
Sensitivity, irritability, a mutual passion for
criticism, and an exaggerated need for recognition alienated the brothers, of
whom Jacob had the slower but deeper intellect.
As suggested by this quote the brothers were equally
at fault in their quarrel. Johann
Bernoulli would have liked the chair of mathematics at Basel which Jacob held
and he certainly resented having to move to Holland in 1695. This was another
factor in the complete breakdown of relations in 1697.
Of course the dispute between the brothers over who
could obtain the greatest recognition was a particularly stupid one in the
sense that both made contributions to mathematics of the very greatest
importance. Whether the rivalry spurred them on to greater things or whether
they might have achieved more had they continued their initial collaboration,
it is impossible to say. We shall now examine some of the major contributions
made by Jacob Bernoulli at an important stage in the development of mathematics
following Leibniz's work on the
calculus.
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 on infinite
series and published his law of large numbers in probability theory. The
interpretation of probability as relativefrequency says that if an experiment
is repeated a large number of times then the relative frequency with which an
event occurs equals the probability of the event. The law of large numbers is a
mathematical interpretation of this result. 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 (1/n) 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 (1/n^{2}) but he did show that it converged
to a finite limit less than 2. Euler
was the first to find the sum of this series in 1737. Bernoulli also studied
the exponential series which came out of examining compound interest.
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 firstorder 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 equation",
y' = p(x)y + q(x)y^{n}
and Hofmann describes this part of his work as:
... proof of Bernoulli's careful and critical work on
older as well as on contemporary contributions to infinitesimal mathematics and
of his perseverance and analytical ability in dealing with special pertinent
problems, even those of a mechanicaldynamic nature.
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.
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. There are
interesting thoughts on what probability really is:
... 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 ...
In Hofmann sums up Jacob Bernoulli's contributions as
follows:
Bernoulli greatly advanced algebra, the infinitesimal
calculus, the calculus of variations,
mechanics, the theory of series, and the theory of probability. He was
selfwilled, obstinate, aggressive, vindictive, beset by feelings of
inferiority, and yet firmly convinced of his own abilities. With these
characteristics, he necessarily had to collide with his similarly disposed
brother. He nevertheless exerted the most lasting influence on the latter.
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.
Jacob Bernoulli continued to hold the chair of
mathematics at Basel until his death in 1705 when the chair was filled by his
brother Johann. Jacob had always found the properties of the logarithmic spiral
to be almost magical and he had requested that it be carved on his tombstone
with the Latin inscription Eadem Mutata Resurgo meaning "I shall arise the
same though changed".
J J O'Connor and E F Robertson
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