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\noindent{\bf Christiaan HUYGENS}\\
b. 14 April 1629 - d. 8 July 1695
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\noindent{\bf Summary.} Trained
to become a diplomat, a career which did not eventuate due to 
political circumstances, Christiaan Huygens turned to science and mathematics. 
While in Paris he heard of the Pascal's and Fermat's attempts to solve gaming 
problems. Back in Holland he conceived his calculus of expectations which 
considerably influenced the following generation of probabilists.
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Christiaan Huygens was born in the Hague on April 14, 1629, the son of the
diplomat, writer, and poet Constantijn Huygens whose Dutch
and Latin verse gained him a lasting place in the history of Dutch literature. 
Because of its services to the house of Orange in two generations the Huygens
family had risen to high social rank.  Christiaan was educated by private
tutors and his father before he went to Leiden in 1645 to study law and
mathematics with the younger Frans van Schooten. He studied classical Greek
mathematics and the new methods of Vi\`{e}te, Descartes, and Fermat (q.v.).

In March 1647 Christiaan Huygens matriculated at the Collegium
Auriacum (Orange College) in Breda, again to study law. He also privately
continued his studies of mathematics.  After his return to the Hague in
August 1649 he went to Denmark as a member of a diplomatic mission in the fall
of 1649. However, after the death of the stadholder William II in 1650, 
Huygens had no chance to enter the
diplomatic service for which he was prepared by his education.
 
The 17 years between the end of his studies in Breda and his departure to
Paris in April 1666 which were spent in the Hague but with short visits
to Paris and
London were the most fertile of Huygens' career. 
He worked from 1650 to 1666 supported by an allowance supplied by his father,
as a gentleman scientist, on  problems of mathematics, mechanics,
astronomy, the construction of pendulum clocks and of optical instruments, the 
lenses of which he ground himself. The discovery of the ring of
Saturn in 1655/56 and the invention of the pendulum
clock in 1656 made him famous and by the mid 60's he was considered 
the leading mathematician and natural philosopher of his time.
 
The following 14 years he spent in Paris 
at the Academy of Science.
He continued his work in mathematics, mechanics,
astronomy and technical problems and added
to his interests fundamental questions concerning the cause of gravity and
the nature of light.

The time in Paris was interrupted by several stays in the Hague where he tried 
to cure an illness which began in 1670. In 1672 the Netherlands were attacked
by Louis XIV. William III, prince of Orange, was appointed Stadholder and
Christiaan Huygens' father together with one of his sons played a considerable
role in the defence against the French. One of the most urgent problems the
Netherlands had to solve in this situation was to recruit soldiers for the 
military defence. The Radspensionaris of Holland, Jan de Witt, had worked 
out a financial arrangement, the Waerdye, according to which annuities on lives
appeared as the most favourable means for the state to raise funds for this
purpose. De Witt's expertise was based on the same principle as Huygens work
on games of chance. Christiaan Huygens was informed of these activities 
but he seemingly
felt no tensions concerning his loyalty. He not only remained in Paris but
also dedicated his book on the mathematical theory of the pendulum clock,
in 1673, to the French king. 

When he again returned to the Hague in September 1681 because of his illness
he decided not to return to Paris where the situation for protestants even
from foreign countries had deteriorated. In the years until his death on
July 8, 1695 he finished a series of works amongst which his
{\it Trait\'{e} de la
lumi\`{e}re} published in 1690 and containing his theory of
light and ideas about gravity, is perhaps the most famous.

Huygens' encounter with the world of stochastics took place quite early in
his career.  During his first stay in France in 1655 he had heard about 
letters exchanged in 1654 between Pascal (q.v.) and Fermat in which the two had
discussed problems concerning games of chance. At that time Huygens
neither met with Pascal or Fermat nor could he gather details about the
methods used in the solution of these gambling problems. 
However, shortly after his return from France Huygens worked out a
method to solve
the problems 
based on his understanding of
``expectation". From a conceptual point of view Huygens' ``expectation" is
different from ``expectation" in later probability theory but both concepts
yield the same values in the cases treated by Huygens.
Huygens did not use the word probability 
in his solutions of chance problems. In his own
understanding the method he applied for these solutions served only the
purpose to demonstrate the power of the new algebra created by
Vi\`{e}te and Descartes. 
The truth of Vi\`{e}te's statement that the new algebra leaves no sensible
problem unsolved could now be demonstrated by Huygens' success to apply
algebra to the realm of chance which hitherto seemed inaccessible for
mathematics.
 When Huygens informed his former teacher Frans van Schooten
of his work on games of chance van Schooten
offered him space for a publication in his forthcoming book.  So Huygens'
tract {\it De ratiociniis in ludo aleae} came out 1657 in Leiden as an
appendix
to Frans van Schooten's {\it Exercitationum Mathematicarum Libri Quinque.}
 
All the problems solved by Pascal, Fermat, and Huygens can be reduced to
two problems, the problem of points and the problem of dice,
both of which can be traced back at least to the late middle ages. 
The problem of points presupposes that a game is not
decided by just one attempt of e.g. throwing a die but by a whole series of
single attempts.  In general the parties involved agree that the winner of
the whole game and by that of all the stakes is the party which first won a
certain number of single games.  If the whole game cannot be finished for
some reason and the parties have to leave before any of them has reached the
necessary number of wins, the stakes have to be divided according to the
number of wins the respective parties are lacking. Since the whole game can
be interrupted at any time, those concerned with the solution of the problem of
points began at least in the 14th century to investigate the new distribution
of the stakes after each single game. So it became interesting to know how
much of the opponent's stake would go to the side of the winner of a single
game, be it the first, the second, and so on up to the last game. 

The problem of dice was formulated in the mid-17th century as asking 
how many throws of a die are needed to get at least n aces.
Huygens based the solution of these problems on the
principle of a just game of chance. 
Participants
engage in a game of chance because they hope for a gain and they know that
they have to pay for this hope with the risk of a loss.
A presupposition for justness is that the sum of the
stakes is equal to the sum of the payoffs of the players, or, in other words,
that there is no third party who
takes a share of the stakes for his service to organize the game. 
Whatever game of
chance is played, its end, that is to say, who won and who is
entitled to the loser's stake is clear by
unambigous rules which were valid centuries before Huygens. Huygens
generalized the situation by admitting that the winner is entitled to a part
of the loser's stake.  This part can be less than the whole stake, but, of
course, it must be positive.
Huygens defined a just game with the postulate, that in gambling
the expectation or share that somebody can claim for something is to be
estimated as much as that with which, having it, he can arrive at the same
expectation or share in a fair game. Huygens' fundamental principle contains
the term expectation which is not explained explicitely. The expectation of
a player A engaged in a game of chance in a certain situation is identified
with his share of the stakes if the game is not played or not continued.
If the game is not played or continued with player A who will be replaced by
a player B, B has to refund A by an amount equal to the expectation of A in
this situation in order to engage in a just game.    Chance was considered by
Huygens and his predecessors as a self evident term. Chance meant for them an
unpredictable and hence uncertain event. In order to subjugate chance to
mathematics it was necessary to select a class of these unpredictable events
characterized by equipossibility which was considered by Huygens as
something elementary and clear.  As paradigms for equipossible cases were
offered the outcomes of throwing a die, tossing a coin, participation in a
lottery, or choosing between two hands hiding different amounts of money.
The more complex problems were solved by reduction to equipossible cases.
Huygens and his predecessors still lacked a problem which forced them to
go beyond this.
But a generation later, with Jakob Bernoulli (q.v.), new cases of unpredictability
like dying in a certain age were taken into consideration which seemed to
be neither equipossible nor reducible to equipossibility. 
For the successors of Huygens it began to matter that a more skilled and able
player would win more frequently than his opponent in a series of games.
Because equipossiblity had not become
problematic to him frequency did not figure in Huygens' tract of 1657.
 
The first
three propositions in Huygens' tract served for the determination of
expectations in concrete situations. The first proposition is that
if it is equally easy to obtain the amount $a$ or $b$ the value of the 
expectation is $(a + b)/2$,
while the second proposition deals similarly with three equally likely
amounts. 
Huygens' central proposition 3 says that if 
the number of cases for gaining $a$ is $p$, and the number of cases 
for gaining $b$ is $q$, then assuming that all cases can happen equally
easily, the expectation is $(pa + qb)/(p + q)$.   The first three
propositions show that
there is no need for Huygens to revert to any notion of probability as
available at the time in order to explain his understanding of expectation or
his way to determine it. Instead Huygens felt a need to demonstrate the first
three propositions of his tract in the most rigorous way.

In all three cases Huygens based his proof on a system of mutually
symmetrical contracts between  players. However, an analysis of the
proofs of propositions 2 and 3 shows that they hold only if the very
meaning of winning a game, and by that, common sense is given up. 
Huygen's proofs allow 
the possibility that the winner
of a game between players who staked the same amount could go away with less
than any of the losers. 

Huygens was very much aware that some would accuse him of support with
his tract for the frivolity of gaming, but he hoped that most of his
readers would
appreciate the utility of his work. Huygens uses the first three propositions
to solve 11 problems in propositions 4 to 14. The first six problems deal with
different situations of the problem of points presupposing equal chances for
the players involved.  Huygens' procedure for the solution of
the problem of points involved establishing a difference equation
for recursive calculation of expectations.
Then, from the situation
of equal chances Huygens proceeded to problems involving unequal chances
for which his paradigm was the throwing of dice.
Typical
is the ninth problem (proposition 12): ``To find how many dice should one take
to throw two sixes at the first throw." Huygens added to the 14 propositions
of his tract five problems the solutions of which he left to his readers.
For the first, third and fifth of these problems which had been posed by
Fermat and Pascal he gave the numerical results without the appropriate
reasoning. This is the content of Huygens' tract from 1657 covering some 20
pages which came out again in the original Dutch wording in 1660. Because of
its enormous impact on the following generation of mathematicians concerned
with stochastic problems it has been called the first book on mathematical
probability. However,it was neither a book - it was published as
a short appendix to a book of Frans van Schooten - nor was it on probability
since neither the word nor a concept of probability were used in the tract.
Huygens' ``theory" of games of chance was the 
intellectual game of a mathematician.

After 1657, Huygens repeatedly returned to problems of chance. In every case
he was induced to these activities by others. He was eager to demonstrate
that his method of calculating expectations by recursion sufficed to solve
problems for
which others used combinatorial methods.  None of
these later stochastic considerations was published; so his later work,
had no
impact on the following generations of mathematicians. 
When John Graunt (q.v.) brought out his {\it Observations Made upon the Bills
of Mortality} in 1662 a copy was sent to Christiaan Huygens. The
{\it Observations}
contained a very short life table and this 
was used by Christiaan to discuss with his brother
Lodewijk in a correspondence from 1669 on the basis of his calculus of
expectations in games of chance and the lottery model the difference of what
is called today the expected and the median lifetime; in addition he
approached the problem of joint-life expectations. The letters exchanged
between the two brothers in 1669 would have been an important source of
inspiration for all who dealt with mortality problems shortly afterwards but
remained unknown until their publication in the Oeuvres in 1895. On the other
hand Huygens' tract of 1657 was, relative to the small number of active
mathematicians at the time, one of the most influential papers in the
history of mathematics.
In England Huygens' tract
appeared in an English translation extended by the combinatorial methods
propagated by Pascal in 1692. Jakob Bernoulli who transformed Huygens' concept
of expectatation and used this to introduce the classical
measure of probability reprinted Huygens' tract together with his annotations
in the first book of his {\it Ars conjectandi} the first book centered around
probability as the main concept of a new mathematical theory.
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\noindent{\bf Bibliography}

\noindent
A good account of Huygens' tract and his other activities in
stochastics even if with the unjustified assignation to probability theory 
is given in:\\
Hald, Anders, (1990). {\it A History of Probability and Statistics and their
Applications Before 1750}, John Wiley \& Sons, Chapter 6.

\noindent
The impact of Huygens' tract is dealt with in:\\
Schneider, Ivo, (1980). The contributions of Christiaan Huygens to 
the development of a calculus of probabilities. {\it Janus}, {\bf 67}, 
269-279.

\noindent
An analysis of Huygens' proofs of the first three theorems is published in:\\
Schneider, Ivo, (1996). Christiaan Huygens' non-probabilistic approach to a
calculus of games of chance. {\it De Zeventiende Eeuw}, {\bf 12}, Nr. 1,
171-185.
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\hfill{Ivo Schneider}

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