Christiaan Huygens

b. The Hague, Netherlands 14 ,April 1629
d. The Hague, 8 July 1695

Huygens made three stays in Paris during his lifetime. The first, to complete his education, lasted from mid-July to November 1655, and was in the company of his brother Ludowijk and his cousin Doublet. This visit permitted Huygens to meet Gassendi, Roberval, Sorbière and Boulliare, the scholars who eventually would found the Académie Royale des Sciences, as well as Claude Mylon, a friend of Roberval and Carcavi.

During a second stay from October, 1660 to March, 1661 he met Pascal, Auzout and Desargues. He then left Paris for London where he attended meetings in Gresham College and met, among others, Sir Robert Moray, one of the founders of the Royal Society of London.

In 1663, he became a member of the Royal Society of London. The next year he was offered membership in a proposed Académie Royale des Sciences based in Paris which he accepted. This society actually began as a set of informal meetings led by Mersenne. The academy was given a charter in 1666. Thus began a third stay in Paris which lasted until 1681.

Van Rekeningh in Spelen van Geluck.

In the history of probability and statistics, Christiaan Huygens is justly famous for he may be credited with having written the first published work on probability.

The Chevalier de Méré posed his two problems regarding games of chance to Blaise Pascal and to Giles Roberval. In their correspondence of 1654, Pascal and Pierre de Fermat were the first to correctly analyze these problems. Since Carcavi served as intermediary between these two men and since Roberval was aware of the problems as well, it would be quite surprising that their work did not become somewhat common knowledge in scientific circles. Thus, although Huygens met neither Pascal nor Fermat during his first visit to Paris, he became aware of the problems but apparently, as is clear from his correspondence, not of their methods.

On his return to Holland in 1655, he began composition of a work on games of chance. In March of 1656 he sent the portion devoted to dicing to van Schooten and, on the 20th of April that same year, a nearly complete manuscript of the treatise known in Dutch as Van Rekeningh in Spelen van Geluck.

Van Schooten offered to append Huygen's treatise to his own Exercitationum Mathematicarum Libri Quinque. As this work was to appear in both Latin and Dutch editions, it was necessary that the treatise be rendered from Dutch into Latin. After some discussion, van Schooten eventually did the translation himself and sent, during March 1657, the Latin version for final additions and corrections. At this time, Huygens added Proposition IX and the Exercises to the treatise.

Appended then to the Exercitationum mathematicarum is the short work De Ratiociniis in Ludo Aleae. This was published in August or September of 1657. Three years later, the Dutch edition Mathematische Oeffeningen to which is appended Van Rekeningh in Spelen van Geluck appeared.

Two translations from Latin into English followed at later dates. An anonymous English translation probably made by Arbuthnot appeared in 1692. Another by a W. Browne appeared in 1714. This version  (Latex or Postscript) can be obtained from University of York. I offer my own of the De Ludo made from the French translation of the Dutch here. In addition, a translation from the Dutch into English by Ernst Willem Scott appears in Papers and Transactions Volume IV (1895-1896) of the Actuarial Society of America, pages 314-328. 

Correspondence of 1656 and 1657.

Evidence that Huygens did not know the methods of Pascal and Fermat lies in this correspondence. Between April 1656 and August 1657, Huygens exchanged a number of letters with Roberval, Mylon and Carcavi in search of confirmation of his methods. This correspondence and the numbers assigned to them appear in the Oeuvres Complètes of Huygens, Tomes I and II.

On 18 April 1656, Letter No. 281, Huygens wrote to Roberval regarding the problem of Proposition XIV to which he requested a solution. Not receiving a response, he appealed to Mylon sometime prior to 13 May with the same problem. This occasioned through the intermediaries Mylon and Carcavi that the problem came to the attention of Fermat and Pascal.

It is, in fact, quite instructive to trace through the correspondence of Huygens with these two men. We find that Huygens is confirmed in his methods. But in addition, he is challenged by new problems from Fermat (Document No. 301) and Pascal (Document No. 336.) The result of these exchanges is that Huygens finds he may make two additions to his treatise. These are Proposition IX which concerns the general problem of points and the collection of exercises placed at the end of the treatise.  The complete set of correspondence of 1656 is available here.

We also include a section from his notebook, Appendix I, bearing on the problem of points which is found in Volume XIV of the collected works and an expression of the general solution to the problem.

Exercises from the Treatise

I. A and B play together with 2 dice to the following condition: A will have won if he throws 6 points, B if he throws 7 of them. A will be the first a single throw; next B 2 successive throws; then anew A 2 throws, and thus in sequence, until the one or the other will have won. One demands the ratio of the chance of A to that of B? Answer: as 10355 is to 12276. Solution.

II. Three players A, B and C take 12 tokens of which 4 are white and 8 black; they play to this condition that the one will win who will have first, in choosing blindly, drawn a white token, and that A will choose first, B next, then C, then anew A and, thus in sequence, in rotation. One demands the ratio of their chances? Solutions.

III. A wagers against B, that of 40 cards, of which 10 of each color, he will draw 4 of them in a way to have one of each color. One finds in this case that the chance of A is to that of B as 1000 is to 8139. Solution.

IV. One takes as above 12 tokens of which 4 white and 8 black. A wagers against B that among 7 tokens that he will draw from them blindly, there will be found 3 white. One demands the ratio of the chance of A to that of B. Solutions.

V. Having taken each 12 tokens, A and B play with 3 dice on this condition that to each throw of 11 points, A must give a token to B, but that B must give 1 to A on each throw of 14 points, and that the one there will win who will be the first in possession of all the tokens. One finds in this case that the chance of A is to that of B as 244140625 is to 282429536481. Solution.

This last problem is known as the Duration of Play.

Correspondence of 1665 with Hudde

Jan Hudde (1628-1704) was introduced to mathematics by Franz van Schooten around 1648. It is interesting to note that three essays by Hudde were likewise incorporated into the Exercitationum mathematicarum of the latter to which was appended the work of Huygens. After 1663, Hudde essentially ceased his mathematical work. However, Hudde and Huygens exchanged various problems and their solutions arising from the treatise of Huygens. This correspondence and the numbers assigned to them appear in the Oeuvres Complètes of Huygens, Tomes V.

The first problem was proposed by Huygens to Hudde 4 April 1665.

A and B play in turn heads or tails, under the condition that the one who casts tails will stake each time a ducat, but that the one who casts heads will take all that which is staked, and A casts first, while nothing has been yet staked. The question is, how much A loses when he enters into the game, or how much he would be able to give to B in order to be able to end it?

Hudde first gives the erroneous solution 1/6. Huygens says that the solution is 4/27. Hudde then gives the solution 2/9. Worked solutions.

The second problem was proposed by Hudde to Huygens in the letter of 5 May 1665

A and B draw blindly in turn. A always 1 of 3 tokens, of which three there are two whites and one black, B likewise always from a certain number of white and black tokens, of which the ratio remains invariable; under the condition that the one who draws a white token will enjoy all that which is set, but that to the contrary the one who draws a black will always add a ducat: and A will draw first before anything has been set. One demands, when one wishes to have some equivalent conditions on both sides, so that A commencing to draw, there is no advantage for any of the two, what ratio should be found between the aforesaid white and black tokens?

Hudde gives the solution that B must have 2 White and 3 Black. Huygens, however, finds that no solution in integers exists. He approximates the ratio of White to Black first as 11 to 7 and as 1193 to 750. This latter ratio is the 8th convergent of the continued fraction expansion of his solution.  This problem is quickly generalized by permitting A to have any ratio of White and Black tokens. Worked solutions.  

The third problem was proposed by Huygens to Hudde on 10 May 1665. If we refer back to the first problem, we find that A is expected to lose money because he has the first turn and there is no money in the pot at the beginning of the game.

A and B cast in turn at Heads or Tails, under the condition that the one who casts tails will set a ducat, but that the one who casts heads will take all which is set; and A will cast first. One demands how much A and B must set from the beginning, that is to say, each an equal sum, in order to make that the condition of A and B become the same?

There is no ambiguity in this problem. Both obtain the same solution of 2/3 ducat. Worked solution.

The last problem was proposed by Huygens to Hudde sometime in 1665.

A and B cast in turn Heads or Tails, under the condition that the one who casts tails will set each time a ducat into the pot, but the one who casts heads will receive each time a ducat if something has been set. And A will cast first when there is yet nothing in the game, and the game will not end before something has been set, and one will play until all has been removed. One demands what is the advantage of A?

Huygens is known to have worked on this problem 15 July 1665 where he obtains the advantage of A to be -1/6 ducat. We also have Item No. 1447 from Hudde which must be dated after 21 August 1665 in which he obtains the equivalent: that B is expected to gain 1/6 ducat. Worked solution.

The complete set of correspondence of 1665 is available and the four appendices of Hudde. 

We have also an isolated letter  No. 1468 from Frenicle de Bessy (1605-1675) to Huygens regarding the game of heads or tails. 

Selections from notebooks found in Volume 14 of Huygen's Collected Works

• Appendix I (pp. 92-95) The solution of Prop. VIII of the text of the Treatise and two problems treated in Prop. IX. It is likely it was composed in 1656 due to correspondence with Carcavy.
• Appendix II (pp. 96-101) Solutions of the 2nd and 4th of the Exercises. It must date from 1665. Its results are included in the letter of Huygens to Hudde of 4 April 1665.
• Appendix III (pp. 102-107)  The solution of the problem posed by Huygens in his letter to Hudde of 10 May 1665. 
• Appendix IV (pp. 108-115) Solution of the problem posed by Hudde in his letter of 5 May 1665.
• Appendix V (pp. 116-150) This is divided into six sections: (1) Solution of the game of heads and tails posed by Hudde in a letter of 4 April 1665; (2) Explication of the calculations reproduced in Appendix III; (3) Explication of the calculations found in Appendix IV. (4) The solution of a problem posed by Huygens in his letter to Hudde of 10 May 1665. (5) The solution of a problem on which Hudde and Huygens are occupied in the months of July and August 1665. (6) How to sum the series composed of reciprocals of the triangular numbers.
• Appendix VI (pp. 151-155) An attempt to justify and generalize the solution of the last of the Exercises in the Treatise.
• Appendix VII (pp. 156-163) Solutions to the problems of the dice related to Propositions X and XI. The notes are undated but it is conjectured that they were written about 1676.
• Appendix VIII (pp. 164-168) On the game of Bassette.
• Appendix IX (pp. 169-178) Work on several problems.  

Further discussion of the problems

• Juan Caramuel, Mathesis Biceps, Kybeia: De Ludus (1670). Caramuel republishes the De ratiociniis in ludo aleae.
• Benedict Spinoza, The ``Reeckening van kanssen" was published anonymously in 1687 at La Hague and is included among this collected works. However, there is some doubt that Spinoza is the author. Whoever was responsible solved only the first problem of Huygens.
• Jakob Bernoulli, Ars Conjectandi, 1713. A recent translation (2006) by Edith Sylla has been published under the title The Art of Conjecturing. Bernoulli analyses the De ratiociniis in ludo aleae completely. See pages 174-192 of this work for his analysis of the problems.
• Montmort, Essai d'Analyse sur les Jeux de Hazard,  1708, 1713. We have here the solutions to all five problems taken from the 2nd edition. Included are comments on the first edition and  solutions offered by both John and Nicolaus Bernoulli.
• Abraham de Moivre, "De Mensura Sortis," 1712; The Doctrine of Chances, 1718, 1738, 1756. De Moivre does not solve the third problem. The first appears as the 13th Problem only in the "De Mensura Sortis." See International Statistical Review / Revue Internationale de Statistique, Vol. 52, No. 3. (Dec., 1984), pp. 229-262. and pp. 246-247 in particular.The 2nd, 4th and 5th are Problems X,  XX and VII respectively in the 3rd edition of  The Doctrine of Chances which is widely available. 
• Nicolas Struyck, Uytreekening der Kansen in het speelen, 1716. Struyck solves all five problems. 

Correspondence of 1662 and 1669: Huygens' on life expectancy

Sir Robert Moray (c. 1608-1673), first president of the Royal Society of London, sent to Christiaan Huygens a copy of Graunt's Observations on the Bills of Mortality shortly after its publication in 1662. This did not elicit much response from Huygens.

Approximately seven years later, Christiaan's brother Ludewijk remarked to Christiaan that he had acquired a copy of Graunt's Observations on the Bills of Mortality.  A correspondence between them ensued over several months concerning the computation of the duration of single and joint lives.

This correspondence together with a few related notes have been collected together. The reference numbers are taken from the Oeuvres Complètes of Christiaan Huygens, Tome VII.

Letters No. 997 and No. 1013 are from Moray to C. Huygens . Letter No. 1022 is the reply of C. Huygens to Moray's two letters.

The latter exchange of letters between Ludwig and Christiaan is listed in the following table.

Three related documents may be examined. These are a copy of Graunt's Table, item No. 1772, an appendix of computations adjoined to Letter 1776, item No. 1777, and a set of notes, a second appendix to Letter 1776, written to himself by Christiaan, item No. 1778.

The entire correspondence on mortality can be obtained here.