Huygens, Christiaan - Encyclopedia of Mathematics (2024)

Christiaan HUYGENS

b. 14 April 1629 - d. 8 July 1695

Summary. Trainedto 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.

Christiaan Huygens was born in the Hague on April 14, 1629, the son of thediplomat, writer, and poet Constantijn Huygens whose Dutchand 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 Huygensfamily had risen to high social rank. Christiaan was educated by privatetutors and his father before he went to Leiden in 1645 to study law andmathematics with the younger Frans van Schooten. He studied classical Greekmathematics and the new methods of Viète, Descartes, and Fermat.

In March 1647 Christiaan Huygens matriculated at the CollegiumAuriacum (Orange College) in Breda, again to study law. He also privatelycontinued his studies of mathematics. After his return to the Hague inAugust 1649 he went to Denmark as a member of a diplomatic mission in the fallof 1649. However, after the death of the stadholder William II in 1650, Huygens had no chance to enter thediplomatic service for which he was prepared by his education.

The 17 years between the end of his studies in Breda and his departure toParis in April 1666 which were spent in the Hague but with short visitsto Paris andLondon 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 ofSaturn in 1655/56 and the invention of the pendulumclock 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 addedto his interests fundamental questions concerning the cause of gravity andthe 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 attackedby Louis XIV. William III, prince of Orange, was appointed Stadholder andChristiaan Huygens' father together with one of his sons played a considerablerole in the defence against the French. One of the most urgent problems theNetherlands 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 livesappeared as the most favourable means for the state to raise funds for thispurpose. De Witt's expertise was based on the same principle as Huygens workon games of chance. Christiaan Huygens was informed of these activities but he seeminglyfelt no tensions concerning his loyalty. He not only remained in Paris butalso 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 illnesshe decided not to return to Paris where the situation for protestants evenfrom foreign countries had deteriorated. In the years until his death onJuly 8, 1695 he finished a series of works amongst which hisTraité de lalumière published in 1690 and containing his theory oflight and ideas about gravity, is perhaps the most famous.

Huygens' encounter with the world of stochastics took place quite early inhis career. During his first stay in France in 1655 he had heard about letters exchanged in 1654 between Pascal and Fermat in which the two haddiscussed problems concerning games of chance. At that time Huygensneither met with Pascal or Fermat nor could he gather details about themethods used in the solution of these gambling problems. However, shortly after his return from France Huygens worked out amethod to solvethe problems based on his understanding of"expectation". From a conceptual point of view Huygens' ``expectation" isdifferent from ``expectation" in later probability theory but both conceptsyield the same values in the cases treated by Huygens.Huygens did not use the word probability in his solutions of chance problems. In his ownunderstanding the method he applied for these solutions served only thepurpose to demonstrate the power of the new algebra created byViète and Descartes. The truth of Viète's statement that the new algebra leaves no sensibleproblem unsolved could now be demonstrated by Huygens' success to applyalgebra to the realm of chance which hitherto seemed inaccessible formathematics.When Huygens informed his former teacher Frans van Schootenof his work on games of chance van Schootenoffered him space for a publication in his forthcoming book. So Huygens'tract "De ratiociniis in ludo aleae came out 1657 in Leiden as anappendixto Frans van Schooten's Exercitationum Mathematicarum Libri Quinque.

All the problems solved by Pascal, Fermat, and Huygens can be reduced totwo 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 notdecided by just one attempt of e.g. throwing a die but by a whole series ofsingle attempts. In general the parties involved agree that the winner ofthe whole game and by that of all the stakes is the party which first won acertain number of single games. If the whole game cannot be finished forsome reason and the parties have to leave before any of them has reached thenecessary number of wins, the stakes have to be divided according to thenumber of wins the respective parties are lacking. Since the whole game canbe interrupted at any time, those concerned with the solution of the problem ofpoints began at least in the 14th century to investigate the new distributionof the stakes after each single game. So it became interesting to know howmuch of the opponent's stake would go to the side of the winner of a singlegame, 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 theprinciple of a just game of chance. Participantsengage in a game of chance because they hope for a gain and they know thatthey have to pay for this hope with the risk of a loss.A presupposition for justness is that the sum of thestakes is equal to the sum of the payoffs of the players, or, in other words,that there is no third party whotakes a share of the stakes for his service to organize the game. Whatever game ofchance is played, its end, that is to say, who won and who isentitled to the loser's stake is clear byunambigous rules which were valid centuries before Huygens. Huygensgeneralized the situation by admitting that the winner is entitled to a partof the loser's stake. This part can be less than the whole stake, but, ofcourse, it must be positive.Huygens defined a just game with the postulate, that in gamblingthe expectation or share that somebody can claim for something is to beestimated as much as that with which, having it, he can arrive at the sameexpectation or share in a fair game. Huygens' fundamental principle containsthe term expectation which is not explained explicitly. The expectation ofa player A engaged in a game of chance in a certain situation is identifiedwith 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 bya player B, B has to refund A by an amount equal to the expectation of A inthis situation in order to engage in a just game. Chance was considered byHuygens and his predecessors as a self evident term. Chance meant for them anunpredictable and hence uncertain event. In order to subjugate chance tomathematics it was necessary to select a class of these unpredictable eventscharacterized by equipossibility which was considered by Huygens assomething elementary and clear. As paradigms for equipossible cases wereoffered the outcomes of throwing a die, tossing a coin, participation in alottery, 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 togo beyond this.But a generation later, with Jakob Bernoulli, new cases of unpredictabilitylike dying in a certain age were taken into consideration which seemed tobe neither equipossible nor reducible to equipossibility. For the successors of Huygens it began to matter that a more skilled and ableplayer would win more frequently than his opponent in a series of games.Because equipossiblity had not becomeproblematic to him frequency did not figure in Huygens' tract of 1657.

The firstthree propositions in Huygens' tract served for the determination ofexpectations in concrete situations. The first proposition is thatif 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 likelyamounts. 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 equallyeasily, the expectation is $(pa + qb)/(p + q)$. The first threepropositions show thatthere is no need for Huygens to revert to any notion of probability asavailable at the time in order to explain his understanding of expectation orhis way to determine it. Instead Huygens felt a need to demonstrate the firstthree propositions of his tract in the most rigorous way.

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

Huygens was very much aware that some would accuse him of support withhis tract for the frivolity of gaming, but he hoped that most of hisreaders wouldappreciate the utility of his work. Huygens uses the first three propositionsto solve 11 problems in propositions 4 to 14. The first six problems deal withdifferent situations of the problem of points presupposing equal chances forthe players involved. Huygens' procedure for the solution ofthe problem of points involved establishing a difference equationfor recursive calculation of expectations.Then, from the situationof equal chances Huygens proceeded to problems involving unequal chancesfor which his paradigm was the throwing of dice.Typicalis the ninth problem (proposition 12): "To find how many dice should one taketo throw two sixes at the first throw." Huygens added to the 14 propositionsof 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 byFermat and Pascal he gave the numerical results without the appropriatereasoning. This is the content of Huygens' tract from 1657 covering some 20pages which came out again in the original Dutch wording in 1660. Because ofits enormous impact on the following generation of mathematicians concernedwith stochastic problems it has been called the first book on mathematicalprobability. However,it was neither a book - it was published asa short appendix to a book of Frans van Schooten - nor was it on probabilitysince 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 casehe was induced to these activities by others. He was eager to demonstratethat his method of calculating expectations by recursion sufficed to solveproblems forwhich others used combinatorial methods. None ofthese later stochastic considerations was published; so his later work,had noimpact on the following generations of mathematicians. When John Graunt brought out his Observations Made upon the Billsof Mortality in 1662 a copy was sent to Christiaan Huygens. TheObservationscontained a very short life table and this was used by Christiaan to discuss with his brotherLodewijk in a correspondence from 1669 on the basis of his calculus ofexpectations in games of chance and the lottery model the difference of whatis called today the expected and the median lifetime; in addition heapproached the problem of joint-life expectations. The letters exchangedbetween the two brothers in 1669 would have been an important source ofinspiration for all who dealt with mortality problems shortly afterwards butremained unknown until their publication in the Oeuvres in 1895. On the otherhand Huygens' tract of 1657 was, relative to the small number of activemathematicians at the time, one of the most influential papers in thehistory of mathematics.In England Huygens' tractappeared in an English translation extended by the combinatorial methodspropagated by Pascal in 1692. Jakob Bernoulli who transformed Huygens' conceptof expectatation and used this to introduce the classicalmeasure of probability reprinted Huygens' tract together with his annotationsin the first book of his Ars conjectandi the first book centered aroundprobability as the main concept of a new mathematical theory.

References

Reprinted with permission fromChristopher Charles Heyde and Eugene William Seneta (Editors),Statisticians of the Centuries, Springer-Verlag Inc., New York, USA.