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\noindent{\bf Nicolaus BERNOULLI}\\
b. 10 October 1687 - d. 29 November 1759
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\noindent{\bf Summary.} This member of the Bernoulli dynasty
was, for a short period in the second decade of the eighteenth
century, the leading figure in all of stochastics, and he has
had a lasting influence.
He edited his uncle's {\it Ars conjectandi} and this has often
been mistakenly regarded as his sole
contribution. He is sometimes confused with a cousin.
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Nicolaus Bernoulli is from the second generation of
mathematicians in the prominent family rooted in Basel,
Switzerland. The tradition had been established by his two
uncles, the famous brothers Jacob (1654--1705) (q.v.) and Johann
(1667--1748), and the second generation is complete with his
three younger cousins Nicolaus (1695--1726), Daniel
(1700--1782) (q.v.) and Johann Jr. (1710--1790) -- sons of Johann.
Also known as Nikolaus, Niklaus, Nicolas or Nicholas (except
for Johannes, we prefer the ``neutral" Latin versions of given
names that the Bernoullis themselves used on many of their
publications), most histories of mathematics mix him with and
mistakenly attribute some of his contributions to his cousin
Nicolaus. While he is
routinely given credit for the editing of the posthumous
publication in 1713 of Jacob's {\it Ars conjectandi},
our Nicolaus'
r\^ole in the development of these subjects was not recognized
until very recently. O.B. Sheynin and
A. Hald pointed out, in 1970 and 1984, the relevance of
his work as a bridge between Jacob's
law of large numbers and de Moivre's corresponding normal
approximation, and subsequently Yushkevich [5] has made a very
strong case for his importance in general. This would not have
been possible without the scholarly publication of the book [1].
However, it was
Hald [4] who analyzed Nicolaus Bernoulli out of obscurity,
presenting him
as an outstanding figure in the early development of
probabilistic and statistical ideas.
This field as a mathematical discipline was new at the time,
born in the correspondence between Pascal (q.v.) and
Fermat (q.v.) in 1654 (the year when Jacob was born),
and it was known from the sixteen pages of Huygens' (q.v.)
{\it De ratiociniis in ludo aleae}
of 1657. Pascal's own
posthumous 1665 booklet, mainly on the combinatorics of the
arithmetical triangle bearing his name, was little known at the
time. As is clear from his {\it Meditationes} published in [1],
Jacob Bernoulli took up studying the subject when he was about
thirty years of age.
Nicolaus Bernoulli was born in 1687 in Basel.
His father Nicolaus (1662--1716), the second son of his
grandfather Nicolaus (1623--1708) between Jacob and Johann,
besides serving on the Basel city council, was an artist: the
known portrait of Jacob, resting his right hand on a globe,
is his painting. The portrait dates from the same
year, 1687, when Jacob occupied the Chair of Mathematics at
the University of Basel as professor.
For the decades of our primary
interest here, Basel was the leading mathematical center of
Continental Europe.
Significantly, 1687 was also
the year of the publication of Newton's
{\it Philosophiae naturalis principia mathematica}. This
culmination of a scientific revolution also marked the
beginning of a transition of everything mathematical. The
Leibnizian version of the Newton$\,$--$\,$Leibniz differential
and integral calculus was in fact championed by Jacob and Johann
Bernoulli; as Leibniz wrote in 1694 to the brothers
(just before the two engaged in vehement quarrels): ``This
method is no less yours than mine." {\it Analysis} (as it has
become known in Continental Europe after l'H\^opital's
influential text of 1696, practically bought in pieces from Johann)
and the combinatorics of Leibniz and Jacob Bernoulli offered
everything that was technically needed for what was becoming
the {\it Art of Conjecturing} in Jacob's hands. It was
perhaps even more important that the success of the
{\it Principia}'s deductive approach for the description of
the ``system of the world" projected great hopes also for
subjects of moral philosophy of the time. To simplify and
exaggerate: Newton has discovered the design of and
created order in the Heavens, so what was left was to do
the same in human affairs on Earth. The mechanics of this
feat was to be probability and, not paradoxically for such
thinking, the philosophical or psychological basis for all
this was the same Protestant determinism -- on the part of
Jacob Bernoulli -- as for Newton for his piece. Even
though the language of probability was necessarily that of
games of chance for the most part at the time, the title
of Part Four of Jacob's {\it Ars conjectandi} promises to
apply the preceding doctrine ``{\it in Civilibus, Moralibus,
\& Oeconomicis.}" The unpublished book stopped unfinished
after the proof of the law of large numbers (the phrase
dating from Poisson (q.v.) in 1837) for an unknown fraction or
probability, illustrated by a numerical example, when its
author, having suffered for some thirteen years from the fever
of what was likely to have been tuberculosis,
died in his fifty-first year.
Nicolaus successfully argued on parts of his uncle Jacob's
work on infinite series for the title of a magister of
arts in 1704. All the Bernoulli mathematicians before and
after him had a comprehensive education, in theology,
law or medicine, as well as in mathematics
and natural philosophy.
Having taken some more mathematical courses
from uncle Johann after 1705, who then took over the
direction of his studies, Nicolaus moved towards law.
Trying to combine it with the {\it Ars} of Jacob, which
he must have known from the manuscript of his original
master, he defended his {\it dissertatio inauguralis
mathematico-juridica} for a doctorate in law at the
University of Basel in 1709, when he was still only 21,
entitled {\it De usu artis conjectandi in jure}. Following
Huygens' and Pascal's booklets from 1657 and 1665 and the
extremely influential first edition of
de Montmort's (q.v.) {\it Essay d'analyse sur les
jeux de hazard} in 1708, the 56-page dissertation is the
fourth bigger {\it published} work on probabilistic
reasoning and, as an effort to try and fulfill at least
part of Jacob's programme, certainly the very first on
``applied" problems; its 12-page summary appeared in the
{\it Acta Eruditorum} in 1711. (The original
dissertation is reprinted in [1], accompanied by K. Kohli's
set of commentaries.)
Nicolaus was unaware of Pascal's book
and from the dating of various correspondence it is also
clear that de Montmort's complimentary copy of his 1708
{\it Essay} to Johann reached Basel only after Nicolaus
defended his thesis in June, 1709.
Giving an overview of his uncle's ideas concerning the
{\it Art of Conjecturing} in general, the dissertation
discusses the calculation from grouped data of mean
and median residual life-time functions, with a clear
distinction between the two, given a survival or
life-time distribution function (at least in points
ten years apart), including joint life expectations;
in particular, he determines the expected value of the
largest of any given number of independent life-times,
uniformly distributed on an interval. The results are
motivated by many examples from civil and canon law
and used for the estimation of the probability that
an absent person is dead, for the determination of the
purchase price of life annuities and the evaluation of
bequests of maintenances, usufructs and life incomes,
of life and marine insurance policies, and of the
expected number of surviving children in the context
of inheritance. Two problems he deals with are different
in nature, but both are related to what will later
become almost an obsession of the author: fairness. One
is the study of fairness in lotteries, where Nicolaus
advises the magistracy to allow profits only for charity
and public good, the other is concerned with ideas about
estimating the credibility of witnesses, suspicions and
testimony in general. As Condorcet's (q.v.)
extensive notes on Nicolaus Bernoulli's thesis ([2])
testify, the
dissertation was rather influential even 75 years after
its appearance. Through Condorcet's published work the
last two problems later influenced Laplace
(q.v.) and particularly Poisson
(q.v.).
Except for the last two problems, things depend crucially
on the underlying survival function. John Graunt's
(q.v.) {\it Natural and Political Observations made
upon the Bills of Mortality} (1662) contained the
first life table, which, however, was {\it ad hoc} in most
part. A 1686 paper of Jacob Bernoulli took over Graunt's
life table from a 1666 resum\'e published in the {\it Journal
des s\c cavans}. Believing that it was based on precise
records of age at death, which it was not, this is what
is used by nephew Nicolaus throughout his dissertation,
though -- unsatisfied with his own assumption of the
uniform distribution of the lifetime in the ten-year periods
of Graunt's life table -- he mentions some Swiss data
collected for him, but unfortunately gives only 18 mean
residual life-times. Problems of this sort were discussed
by the Huygens brothers, Jan de Witt, Jan Hudde,
Edmond Halley and others before
him, particularly the pricing of annuities, but, except for
Halley's paper, these were not published.
Uncle Johann had his best praise about his nephew's talent
and work in his letters to Leibniz. During Nicolaus' shorter
trip to Paris in 1709 and his ``grand tour" to France,
England, The Netherlands and France again in 1712--1713,
the doors of most mathematicians and scientists were open
to him; among many others he met de Montmort (q.v.) in
France, de Moivre (q.v.), Halley and Newton
in London and Willem 'sGravesande in the
Hague. Also, Johann introduced him to correspondence
with several mathematicians. This is how Nicolaus began to
exchange letters with de Montmort, which turned out to be
very significant for our story: Johann, commenting on many
problems in the 1708 {\it Essay} (pp. 283--298 in [3]),
enclosed in his letter of March 17, 1710, Nicolaus' notes
on de Montmort's famous matching problem (pp. 299--303 in
[3]). Indeed, most of what we know about Nicolaus
Bernoulli's work in stochastics besides his dissertation
is from the seven letters of the published part of his
correspondence with de Montmort up to 1713, included in the
second edition of the {\it Essay} [3], pp. 299--412, from
his correspondence with 'sGravesande in 1712, occupying 16
pages in the latter's {\it Oeuvres} published only in 1774
(these two sets are brilliantly analyzed by Hald [4]), from
short excerpts of his later correspondence with de Montmort,
Cramer and his cousin Daniel Bernoulli (q.v.),
appearing in [1], and a few
more letters of his correspondence with Leibniz and Euler
published in other collections. At least 450 more letters
to and from him, owned by the University of Basel, still
remain unpublished.
The debut with de Montmort was spectacular. In February
1711, well within three months after receiving de Montmort's
encouragement to do it, Nicolaus gave a complete solution to the
latter's problem on the duration of play, ``the most difficult
topic in probability theory before 1750" according to Hald [4],
which was approached less completely by de Moivre several
times later. This is the determination of the probability
that in the gambler's ruin problem (with both players having
an arbitrary, possibly different number of ducats to play with
and arbitrary winning probabilities $p$ and $q$, $p+q = 1$, at
each game) the play ends with one of the players ruined in at
most $n$ games. The usual formulae for eventual ruin, stated
without proof by Jacob in his {\it Ars} and proved in
de Moivre's {\it De mensura sortis} in 1712, follow as
$n\to\infty$.
The next problem was what an Englishman, a certain
Mr. Waldegrave, posed to both de Montmort and de Moivre:
players ${\rm P}_1,\ldots, {\rm P}_n$ of equal skill play a
circular tournament, in which first ${\rm P}_1$ and ${\rm P}_2$
play, then the winner plays ${\rm P}_3$, the winner of this match
plays ${\rm P}_4$, and so on, ${\rm P}_1,\ldots, {\rm P}_{n-1}$
entering again after ${\rm P}_n$ if necessary, until someone
beats everyone else in a row. The problem is to calculate the
probability of winning for each player, with the expected payoff
if each loser pays a crown and the winner takes all, and the
probability that the tournament ends on a given number of games.
De Montmort and de Moivre could do this for $n=3$ and $n=4$,
respectively, Nicolaus gave the solution in its
complete generality also in 1711 and thought that this was his
best contribution up to that point. This is the rare exception
when the solution was also published, in the {\it Philosophical
Transactions}, besides [3]: Nicolaus discussed it with de Moivre
in 1712 while in London and subsequently sent a Latin version
to him as well, and he arranged for its publication. Nicolaus
and de Montmort then corresponded about numerous other games,
such as the one named {\it Her}, which is historically the first
example of solving a strategic game of chance.
Nicolaus' following contribution is what posterity probably
takes as his best overall. It is in a letter again to
de Montmort, dated on January 23, 1713 in Paris, on his way
back from London (pp. 388--392 in [3], Russian and English
translations in [5]; a version was also sent to 'sGravesande
earlier). Improving on Jacob's proof for the law of large
numbers for a relative frequency, he gave a large-sample
approximation to a lower bound of the probability that a
binomially distributed count of ``fertile cases" does not
deviate from its integer mean more than a given integer limit.
The lower bound is expressed in terms of ratios of the middle
and corresponding extreme terms of the binomial distribution
and the approximation comes close to the local normal
approximation established by de Moivre twenty years later. He
was motivated by a statistical question: John Arbuthnot
(q.v.) extended Graunt's data on the christenings in
London and used the excess of boys in all the 82 years
1629--1710 for his ``Argument for Divine Providence $\ldots$,"
published in 1712 in the {\it Philosophical Transactions},
concluding that ``it is Art, not Chance, that governs." This
was a hot topic while Nicolaus was in London and he discussed
it with 'sGravesande in the Hague, who improved Arbuthnot's
argument for testing (and rejecting) $p={1\over 2}$, also in
1712, for the probability $p$ that a newborn is a boy. Using
his approximate bound, Bernoulli, giving separate
considerations to the middle and extreme portions of the
data, fits a binomial distribution to the observed numbers
and, taking $p={18\over 35}$, concludes
that ``there is no ground to be surprised that the number of
infants of the two sexes do not differ more [than observed],
which I wanted to show." Misunderstood, as if he wanted to go
against the theological conclusion of Arbuthnot, his findings
were criticized as late as in the third edition of de Moivre's
{\it Doctrine of Chances} in 1756, pp. 252--253.
The kind of data analysis
Nicolaus Bernoulli performed was not repeated until much
later works by Daniel Bernoulli and Laplace.
From Paris Nicolaus went to de Montmort's country estate and
stayed with him for about two months to help him prepare the
second edition [3] of his {\it Essay}. (It would be a joint
book by today's standards; both editions were published
anonymously, but everyone knew who the author was.) He just
got home to Basel, in April 1713, to write a preface for the
publication of the {\it Ars conjectandi} and to include a
list of the printer's errata, and it seems that was all he
did, or was allowed to do by Jacob's widow and son, for the
edition. (Neither the printer nor Jacob's son,
arranging for the publication, was a mathematician.
And to add to the Nicolaus confusion, this cousin of our Nicolaus,
Jacob's son, and a painter, was also Nicolaus (1687--1769), born in the same
year, and known as ``Nicolaus the younger"
to distinguish him from his uncle, our Nicolaus' father,
``Nicolaus the elder.") The book came out in August 1713,
three months before de Montmort's [3]; it could have been
a point that this should happen so.
At this juncture, we see Nicolaus Bernoulli as {\it the}
leading figure of the three main players (de Montmort,
de Moivre and himself) in what Hald [4] describes as
``the great leap forward" in stochastics from the
publication of the first edition of de Montmort's book in
1708 (greatly influenced by the 1706 reviews of Jacob's
unpublished book in the {\it Journal des s\c cavans}) to
the first edition of de Moivre's book
{\it Doctrine of Chances} in 1718. The latter was an
extended and greatly improved English version of the 1712
{\it De mensura sortis}. Although de Moivre was
years Nicolaus' senior by twenty years, and an accomplished mathematician
by that time, he was a relatively late-comer to stochastics
in 1712 (and already 66 years old when he found the normal
approximation later in 1733). Nicolaus' help and leading
r\^ole is fully acknowledged by de Montmort ([3], p. 400).
He was elected to the Berlin Academy
in May 1713 and to the Royal Society of London
a year later. And, mysteriously enough, on this
early zenith of his career the 26-year old leader of
stochastics virtually quit his researches in his field of
prominence.
He waited for three years in Basel for a job opportunity and,
backed by Leibniz, became the professor of mathematics in
Padua in 1716. A
Swiss Protestant, probably not liking the Italian Catholicism
surrounding him there, as Yushkevich [5] surmises, he returned
to Basel in 1719, first occupied the chair of logic
at the University of Basel in 1722 and then became the
professor {\it utriusque iuris} (of both Roman and canon laws)
in 1731, being well respected in that position to the end of his
life. He did publish some good works in
Italian journals and in the {\it Acta Eruditorum} on
differential equations before 1721 and led a correspondence
with Euler on infinite series in the years 1742--1745.
He also tried to continue the correspondence with de Montmort
after 1713 on some games of chance, but de Montmort, maybe
bored by his admitted r\^ole of a second fiddler in this,
was decreasingly interested and, before coming close to
completing his next big project of writing a history of
mathematics, died of smallpox in 1719. However, the
enthusiasm and the quality of these activities of Bernoulli
are on a much lower level than those of his great period
of stochastics for the remaining 46 years of his life.
The real explanation for his leaving stochastics must come
from the state of
stochastics itself. As Nicolaus saw it, with the publication
of the {\it Ars conjectandi} and the {\it Essay} in 1713 and
de Moivre's {\it De mensura sortis} in 1712, what was left was
really just Jacob's programme itself. There would have
been small but inconvenient problems if someone wanted to
give a try starting it. For example, Nicolaus no doubt learned
in London that there was a better life table published by
Halley in the {\it Philosophical Transactions} as early as 1694.
Strictly speaking, his dissertation should have been rewritten
with the moral that it was difficult to get quality data. (Had
he known the first edition of de Montmort's {\it Essay} in 1709,
he would have found a laudatory reference to Halley's paper.)
The greater problem was that no significant concrete problems
were coming forward {\it in Civilibus, Moralibus, \&
Oeconomicis}; and anyway, the statistical tools would have been missing
even to treat them. Jacob's programme, or dream rather,
was wholly impossible to accomplish in the eighteenth century.
It is impossible today, and will remain so, in the sense that
it was understood then. Indeed, it is almost comical how the
programme appears in the prefaces of the
books. De Montmort described this programme in his preface
in 1708 from the eulogies of Jacob and concluded that he
``will not have a Part Four" in his work and will ``leave it
to another person more capable than me." Nicolaus Bernoulli,
in his preface to the {\it Ars conjectandi} wrote in 1713
that he was too young and inexperienced for the task of
completing Jacob's Part Four and that, knowing that
de Montmort would reprint his own preface in the second
edition within a few months, he would rather ask de Montmort
and de Moivre ``to take the task on themselves." Finally,
de Moivre threw the ball back in the preface of his
{\it Doctrine of Chances} in 1718: ``I wish I were capable
of carrying on a Project [Jacob Bernoulli] had begun, of
applying the Doctrine of Chances to {\it Oeconomical} and
{\it Political} Uses, to which I have been invited, together
with Mr. {\it de Montmort}, by Mr. {\it Nicholas Bernoully}:
$\ldots$ but I willingly resign my share of that task into
better Hands, wishing that either he himself would prosecute
that Design, $\ldots$ or that his Uncle,
Mr. {\it John Bernoully}, Brother to the Deceased, could be
prevailed upon to bestow some of his Thoughts upon it."
So, Nicolaus' original idea of ``carrying on the project"
hardly appeared possible in 1713. To complicate matters
further, he accidentally invented a problem in the same year
(again in a letter to de Montmort; [3], pp. 401--402) that
shook his and others confidence in what appeared to be
the basic notion since Huygens: expectation. It was about
Paul's fair price for the game in which Peter gives him
$1, 2, 4, 8, 16, \ldots$ ducats if he, flipping a fair coin,
gets the first `head' on the first, second, third, fourth,
fifth, ... trial. The price is an infinite number of ducats
according to Paul's infinite expectation, but ``any even
half-way sensible person would happily sell his chance for
twenty ducats," as Nicolaus put it in 1728. Expectation to
him was the mathematical version of justice and equity,
the exact foundation on which fairness in human affairs could
and would be built, and this is exactly what he managed to
blow up. From here on, his work in stochastics is virtually
restricted to correspondence on this paradox, known from
1768 as the St.$\,$Petersburg problem,
trying to convince people that it was important and
criticizing their work if he was successful. This is what
happened in the case of Cramer and his cousin Daniel
in the period from 1728
to 1732; the simplified statement of the problem above is
due to Cramer, Nicolaus' original formulation was for
throwing a die until the first `six' appears. While the
well-known utility approach of Cramer and Daniel Bernoulli
has enjoyed an extremely distinguished career in economic
theories later, their ``moral expectations" never satisfied
Nicolaus. He wanted to see a mathematics problem solved,
but it was so
difficult that it had to wait 242 years until stochastics
got mature enough to afford it a reasonable initial treatment,
a rough first approximation by a law of large numbers of
Feller in 1945.
It is his early contributions in his great five years
1709--1713 to a reduced version of Jacob's programme of
stochastics, probability and statistics, for which Nicolaus
Bernoulli is remembered today.
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\begin{thebibliography}{3}
\bibitem{1} [1] {\it Die Werke von Jakob Bernoulli}, Band III.
By the Naturforschenden Gesellschaft in Basel, B.L. van der
Waerden, ed. Birkh\"auser, Basel, 1975.
\bibitem{2} [2] Condorcet (1994). {\it Arithm\'etique politique. Textes rares
ou in\'edits (1767--1789)}, B. Bru et P. Cr\'epel, eds.,
Presses Universitaires de France, Paris.
[3] de Montmort, Pierre R\'emond (1713). {\it Essay
d'analyse sur les jeux de hazard.} Seconde edition, Rev\^ue
\& augment\'ee de plusieurs Lettres. Jacque Quillau, Paris.
{\bf [}Third edition: reprint of the second by Chelsea,
New York, 1980.{\bf ]}
\bibitem{3} [4] Hald, Anders (1990). {\it A History of Probability and
Statistics and their Applications before 1750.} John Wiley,
New York.
\bibitem{4} [5] Yushkevich, A. P. (1986). Nicholas Bernoulli and
the publication of James Bernoulli's {\it Ars conjectandi}
(in Russian). {\it Teoriya Veroyatnoste\u{\i} i ee Primeneniya},
{\bf 31}, 333-352. {\bf [}English translation:
{\it Theory of Probability and its Applications} {\bf 31},
286--303.{\bf ]}
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\hfill{S\'andor Cs\"org\H{o}}
\end{thebibliography}
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