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 \noindent{\bf Jakob BERNOULLI}\\
b. 6 January 1655 - d. 16 August 1705
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\noindent{\bf Summary.} Jacob
Bernoulli, together with his brother Johann one of the pioneers of the
Leibnizian form of the calculus, transformed Huygens' calculus of expectations
to make probability its main concept. He formulated and proved the weak law of
large numbers, the cornerstone of modern probability and statistics.
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Bernoulli stemmed from a family of merchants. His grandfather Jakob
Bernoulli became a citizen of Basel in 1622 by marriage. His father, Nikolaus
Bernoulli, took over the drug business from his father and became a member of
the town council.  After finishing the master of arts in 1671 Jakob Bernoulli
studied theology until 1676 when he received the licentiate in theology; at
the same time he studied mathematics and astronomy secretly against the will
of his father. After 1676 he left Basel to work as a tutor for four
years which he spent in Geneva and France. He also travelled to France,
the Netherlands, England and Germany in 1681, 1682.
His first publication,in 1681, dealt with his observations of the comet
of 1680 and his prediction of its reappearance in 1719.
Back in Basel he began to give
private lectures especially on the mechanics of solid and liquid bodies
and he became interested in analysis.
Two years after its foundation in 1682
Leibniz had published in the Acta Eruditorum a method to determine integrals
of algebraic functions, a short presentation of the differential calculus in
algorithmic form and some remarks concerning the fundamental ideas of the
integral calculus. These papers occupied the interest of Jakob and
his brother Johann
Bernoulli. Jakob tried to get further information from Leibniz in 1687 but he
answered Jakob's questions only three years later because of a
journey  undertaken in diplomatic mission. At that time Jakob and
Johann had not only mastered the Leibnizian calculus but also had
added so considerably to it that Leibniz wrote in a letter of 1694 that the
infinitesimal calculus owes as much to the Bernoulli brothers as to himself.

Jakob also cultivated the theory of series which were published in five
dissertations between 1689 and 1704.  He considered series
as the universal means to integrate arbitrary functions, to square and rectify
curves. In 1690 Jakob had introduced the term ``integral" in his solution of
the problem to determine the curve of constant descent. In
the 90's the relationship between Jakob and Johann deteriorated mainly
because of the hot and excitable temper of the very ambitious younger Johann.
The bitter quarrels between them became as notorious as the priority
dispute between Leibniz and Newton, which began at about the
same time, concerning the creation of the infinitesimal calculus.

In 1687 Jakob
became professor of mathematics at the university of Basel, in which position
he remained until his death in 1705. He was honoured by the memberships of
the Acad\'{e}mie Royale des Science (1699) and of the Academy of Sciences in
Berlin (1701). He had a daughter and a son from Judith Stupan whom he had
married in 1684. His most famous monograph is the {\it Ars
conjectandi} (or Art of Conjecturing) which was only published after his
death in 1713.
This represents the transition from the
calculus of chances as developed by Christiaan Huygens (q.v.) in 1657 to a new
mathematical theory centered around the concept of probability the classical
measure of which was also introduced by Jakob Bernoulli.

The {\it Ars conjectandi},
generally considered as Bernoulli's most original work, remained incomplete.
It consists of four parts. The first is a reprint of Huygens'
{\it De ratiociniis in aleae ludo}, which was published
in 1657, complemented
by extensive annotations which contained important modifications and
generalizations preparing something new, the theory of probability theory. In
the second part Bernoulli deals with the theory of combinations.
In the third part Bernoulli gives
twenty-four examples concerning the determination of the modified Huygenian
concept of expectation in various games. The fourth part is the most
interesting and original. At the same time it is the part which Bernoulli
was not able to complete. In the first three of its five chapters it deals
with the new central concept, probability,
its relation to certainty, necessity and chance, and ways to estimate and
measure it. In chapter 4 Bernoulli distinguishes two ways to
determine exactly or approximately the classical measure of probability. The
first presupposes equipossibility of the outcomes of certain elementary
events like drawing either one of $n$ balls numbered from 1 to $n$ out of an
urn. So the probability to draw a ball of a certain colour out of an urn
filled with balls of different colours is determined
{\it a priori} by the ratio
of the number of balls of this special colour to the number of all balls in
the urn. For the determination of the probability of an event like dying of
a certain person within the next ten years a reduction to numbers of
equipossible cases which are favourable or unfavourable for the event is
impossible. But according to Bernoulli we can inductively, by experiments,
or {\it a posteriori} in his sense,
get as close as we desire to the true measure of such a probability. The
possibility to estimate the unknown probability of such an event by the
relative frequency of the outcome of this event in a series of supposedly
independent trials is secured according to Jakob Bernoulli by his theorema
aureum, which was called later, by Poisson, Bernoulli's law of large numbers.
The proof of this theorem is contained in chapter 5. In an appendix Jakob
Bernoulli treats the jeu de paume, a ball game usually considered as a
predecessor of tennis, as a game of chance.

The idea to give his book the
title {\it Ars conjectandi} was stimulated by the {\it Ars Cogitandi},
better known as
the Logic of Port Royal, in the very last chapter of which the chances for
future contingent events are equated with the ratios of the associated
degrees of probability. One can see how Bernoulli, beginning from this
notion, developed the classical concept of probability,
and how he established as its measure a generalization
of Huygens' expectation, namely, the ratio of favorable to
all possible cases. At the same time he became the first to set down the
prerequisites for consciously formulating a program for the mathematization
of all the fields of application subject to ``probabilis". Bernoulli himself
sought to execute this program, but his premature death prevented him.

Jakob's interest in stochastics began in the 1680's. His first publication
on the calculus of chances dates from 1685. The development of his
ideas can at least in part
be traced in the Meditationes, his so-called scientific diary, which he had
begun in 1677. It shows the significance of jurisprudence for the transition
from a calculus of chance to a calculus of probability. This is accompanied
by Leibniz independent and lifelong interest in a doctrine or a logic of
degrees of probabilities triggered by the hope to quantify conditional rights
but also in the dissertation of Niklaus Bernoulli, Jakob's nephew, who sought
to apply the findings of his uncle to a series of concrete problems in law.
Different from Jakob Bernoulli Leibniz never worked out his ideas in this
field. He left only a series of partly redundant drafts and manuscripts but
no publication. Jakob Bernoulli
learned only late in his life of Leibniz' interest in a doctrine of
probabilities. This is testified by the correspondence between the two men
from April 1703 until Jakob's death in 1705.
Jakob Bernoulli took Leibniz' objections, especially
against the significance of his law of large numbers, as representative of a
critical reader and tried to refute them in part 4 of the
{\it Ars conjectandi}.
Leibniz emphasized that
in the area of jurisprudence and politics, which was so important for
Bernoulli's program, no extended calculations were usually required,
since an enumeration of the relevant conditions would suffice. Against the
possibility
of attaining a better approximation to a sought after probability
with an increasing number of observations, Leibniz suggested that
contingent events, here identified with dependence on infinitely many
conditions, could not be determined by a finite number of experiments.
In addition, e. g., the
appearance of new diseases could change the probability of survival of a
person with a certain age.
However, Bernoulli saw no problem with applying an urn model
to human mortality, with the stones corresponding to diseases with which a
person can be taken ill. So Bernoulli's research program stood firm after his
discovery of the law of large numbers in 1689.

The content of the {\it Ars
conjectandi}  in its most important sections went back to preliminary studies
he had done in the 1680's as testified by the Meditationes. A key passage for
the transformation of Huygens' expectation to probability involves the
treatment of a problem of law, on which Bernoulli worked in 1685/86. It has
to do with a marriage contract, which, assuming that the couple have
children and that the wife (Caja) dies before the husband (Titius), will
govern the
division of their common property between the father and the children.
The portion of the groom will be
larger if he has already entered into his inheritance, smaller if not,
unless both fathers have died. The bride's father objects to his initial
proposal; this induces Titius to make a second proposal, according to which
he will receive
the same portion of the common property regardless of what happens to the
fathers. On this basis Bernoulli poses the question: which suggestion would
be more favorable for the children? To this end he has to make assumptions
about the possible order of death of the three people involved, the two
fathers and Caja. He first assumes that all six possible orders
have equal weight. But this assumption does not satisfy him, since the youth
of the bride has not been taken into account. Thus, he assumes that for every
two instances
e.g., diseases, symptoms or events which might bring about the death
of either father, there is only one which threatens Caja with death. There are
thus five cases in all, each equally likely to take its victim first. Since
Caja is affected by only one of these, while the two fathers are affected by
four, her situation is evaluated as one-fifth of certainty of her being first
to die, that is, ``one probability, five of which make the entire certainty".
Here Bernoulli uses the plural ``probabilities", where these are equated with
the no more precisely distinguished individual cases; this usage does not
permit the conception of ``probability" as ``degree of certainty" which is
observed in the next stage. Aided by Huygens' formula for determining
expectation, Bernoulli then derives a certainty of 4/15, written 4/15 c,
where ``c" stands for ``certitudo", or 4 probabilities our of 15, that Caja
will die second, and finally 8/15 c that she will die third.
At the end
of his treatment of the marriage contract he speaks of the
degree of probability:
\begin{quote}
``Generally in civic and moral affaires things are
to be understood, in which we of course know that the one thing is more
probable, better or more advisable than another; but by what degree of
probability or goodness they exceed others we determine only according to
probability, not exactly. The surest way of estimating probabilities in these
cases is not a priori, that is by cause, but a posteriori, that is, from the
frequently observed event in similar examples."
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Bernoulli carried out the
determination of probabilities a posteriori by adopting relative frequencies
determined through observation as estimates of probabilities which could not
be given a priori.
He felt justified to proceed in this way by his fundamental theorem which at
the same time served as the essential foundation of his program to
extend the realm of application of numerically determinable probabilities.

Bernoulli's understanding of chance excluded events as
occurring indeterminately. He was convinced that through a more precise
knowledge of the parameters affecting the motion of a die, for instance, it
would be possible to specify the result of the throw in advance. In similar
fashion he viewed changes in weather as a determinate process, just as the
occurrences of astronomical events are. Chance, in his view and later in the
view of Laplace, was reduced to a subjective lack of information. Thus,
depending on the state of their information, an event may be described by
one person as chance, but by another as necessary. With this anticipation of
Laplacian determinism Bernoulli appears to solve the problem of the
connection between chance and divine Providence. The entire realm of events
which are described in daily life as uncertain or contingent in their
outcome is such,
he claims, merely because of incomplete information: nevertheless, these too
fall within the field of the concept of ``probabilitas". Bernoulli's program
to mathematize as much of this realm as possible with the aid of the classical
measure of probability occupied researchers like de Moivre (q.v.)
and Laplace (q.v.)
throughout the 18th century and into the second half of the 19th.
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\noindent{\bf Bibliography}

A good account of Jakob Bernoulli's contribution to
probability theory together with an appreciation of the mathematical
methods used is
given in:

\noindent Hald, Anders, (1990). {\it A History of Probability and Statistics
and their Applications Before 1750}, Wiley, Chapters
15 and 16.
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For Jakob Bernoulli's understanding of probability and chance see:

\noindent Hacking, Ian, (1975). {\it The Emergence of Probability}, Cambridge UP,
Chapters 16 and 17 and

\noindent Shafer, Glenn, (1978). Non-additive probabilities in the work of Bernoulli and
Lambert. {\it Archive for History of Exact Sciences}, {\bf 19}, 309-370,
especially pp. 323-341.
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The role of Jakob Bernoulli as the founder of a
mathematical probability theory and his relationship to Leibniz is
treated in:

\noindent Schneider, Ivo, (1981). Why do we find the origin of a calculus of probabilities
in the seventeenth century? In: {\it Probabilistic Thinking, Thermodynamics,
and the Interaction of the History and Philosophy of Science} (Eds. J.
Hintikka, D. Gruender and E. Agazzi) Vol. II, Dordrecht: Reidel
(= {\it Synthese
Library}, Bd. 146), pp. 3-24.

\noindent Schneider, Ivo, (1981). Leibniz on the Probable. In:
{\it Mathematical Perspectives, Essays on Mathematics and its Historical
Development} (Ed. Joseph W. Dauben), New York: Academic Press,
pp. 201-219.

\noindent Ivo Schneider, (1984). The role of Leibniz and of Jakob Bernoulli for
the development of probability theory. {\it  LLULL, Boletin de la Sociedad
Espa\~{n}ola de Historia de las Ciencias}, {\bf 7}, 68-89.
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Very useful
information concerning the background of Bernoulli's work in stochastics is
contained in the commentaries by K. Kohli and B. L. van der Waerden in:
{\it Die Werke von Jakob Bernoulli}, vol. III (ed. by the Naturforschende
Gesellschaft in Basel), Birkh\"{a}user Verlag Basel 1975.
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\hfill{Ivo Schneider}


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