Introduction

Remarks on the texts

Chr. Huygens, 1656-1657, 1660; p 485-500

Huygens' language can be described as clear and simple. The spelling is rather fluid, euphemistically speaking (e.g. oogen and ooghen), but the text as a whole is to the point and unambiguous. There is some variation in the phrasing of the propositions that might be deemed unnecessary according to modern standards of uniformity. In some of the translations (eg the French translation) these small variations are often smoothed away.
Notice that in general Huygens' kans is best translated as occasion, and not as probability
Incorrect references to Proposition II in stead of Proposition III, that occur in Propositions X to XIII, suggest that Proposition II had been ommitted for some time, only to be put back at the last moment. This must have been before Proposition XIV was entered, because it contains two correct references to Proposition II.
Of the two errors in 1657 in mathematical expressions in Proposition III only the second one remained: in the text it still says qx+bx-bq-ap-a in stead of px+qx-bq-ap-a.
In Proposition IV the order of the first two paragraphs is correct. In Proposition IX however the order of the first two paragraphs 'Om tusschen...' and 'Zy genomen...' seems to be reversed, when compared with the other propositions.

F. van Schooten, 1656-1657; p 517-534

The Latin translation by Frans van Schooten was not very highly regarded by Huygens (see letter july 1657). Van Schooten was from the start (april 1656) quite persistent about him translating Huygens' Dutch text while almost completely disregarding Huygens own draft for a Latin translation. The final product is rather tortuous at times but at the positive side it did launch expectation as a mathematical term. At the downside it completely failed to name the concept of probability.
The error in the order of the first two paragraphs of Proposition IV was already noted by Huygens in letter to Van Schooten of september 28, 1657 He also detected the 'horum' which should have been 'hae' in the first paragraph of Proposition IX.
Incorrect references to Proposition II in stead of Proposition III, that occur in Propositions X to XIII, suggest that Proposition II had been ommitted for some time, only to be put back at the last moment. This must have been before Proposition XIV was entered, because it contains two correct references to Proposition II.
Two errors in mathematical expressions in Proposition III: in the text of the proposition itself it says (pa+pb)/(p+q) in stead of (pa+qb)/(p+q), as Stigler in 1992 already noted in his Apollo Mathematicus: "the first typographical error in the history of mathematical probability"; in the text it says qx+bx-bq-ap-a in stead of px+qx-bq-ap-a.
In Proposition IX the order of the first two paragraphs 'Ut tot...' and 'Ponamus...' seems to be reversed, when compared with the other propositions.


J. Arbuthnot, 1692; p 5-48

John Arbuthnot (1667-1735) read An Argument for Divine Providence, taken from the constant Regularity observed in the Births of both Sexes in 1710 and wrote Essay Concerning the Effect of the Air on Human Bodies (1733). The 1st edition of Of Laws...was followed by the 2nd (1714) and 3rd (1717) edition and in 1738 a 4th edition, revised by John Ham, was issued.
The first mishap which occurs at proposition IV (page 15), follows the error of the Latin 1657 edition: the text of this proposition and that of the next paragraph are exchanged (see picture on the left). So the text of the proposition is now set in the font of the normal text, while the text of the paragraph that originally followed, is now italicized like the other propositions.
The next mistake occurs in Problem V where ...that if 12 comes up... should have been ...that if 11 comes up.... As this changes the problem it seems important enough to mention.


Bernoulli, 1713; p 1-71

The text has the order of the first two paragraphs of proposition IV corrected as well as some other errors (like the mathematical expression in Proposition III). The incorrect references to Proposition II in stead of Proposition III, that occur in Propositions X to XIII, are still there. There are some differences in using accents on vowels.

W. Browne, 1714; p 2-24

The translator W. Browne is generally identified with Sir William Browne (3 jan 1692 - 1774) who was an eminent physician, studied medicine at Cambridge and was elected President of the Royal College of Physicians in 1765. He kept a commonplace book from 1708 until 1774.
Richard Mead (11 Aug. 1673-16 Feb. 1754) to whom Browne dedicated his translation, wrote De Imperio Solis ac Lunae in Corpora Humana, et morbis inde oriundis (A Treatise Concerning the Influence of the Sun and the Moon on Human Bodies) (1704). It is amply referred to by Brown in the dedication, who suggested that Medicine could benefit from the calculus of probabilities.
In general Browne is succesful in translating the Latin text by Van Schooten. In at least three instances however he trips and leaves the reader puzzled.
The first mishap which occurs at proposition IV (page 7), follows the error of the Latin 1657 edition: the text of this proposition and that of the next paragraph are exchanged (see picture on the left). So the text of the proposition is now set in the font of the normal text, while the text of the paragraph that originally followed, is now italicized like the other propositions.
The second mishap occurs in the text of proposition XIII and is a matter of translation. In comparison with the other texts it appears that Brown completely misses the point of the second half of the text starting from ..and after this Bargain made..
The third mistake occurs in Problem IV where ...3 black ones... should have been ...3 white ones.... As this changes the problem it seems important enough to mention.


D.J. Korteweg, 1920; p 62-90

Volume 14 of Oeuvres Complètes de Christiaan Huygens published in 1920 contains Huygens' writings on probability. After an 'Avertissement' (page 4-48) the [unnumbered] pages 50-53 show the front pages & the title pages of the fifth book of the Latin (left page) and Dutch (right page) editions of Van Schootens publication. The [unnumbered] pages 54 en 55 show on the right page a Dutch text starting with 'Tot den Leser' and on the left page a French text starting with 'Au Lecteur'. Korteweg explains on page 6 'Ajoutons que celui-ci n'était pas tout-à-fait satisfait de cette traduction; ce qui fut pour nous une raison de plus préférer pour notre texte la version hollandaise à la version latine, quoique cette dernière eùt paru trois années plutôt que l'autre'.
The question rises: what does the French text represent? The layout of the earlier (front) pages suggests that the French text might be a continuation / translation of the Latin text. And I believe that several later authors such as David and and to a lesser extent Dupont & Roero have been duped by this suggestion. The French text is in reality a translation of the facing Dutch text as a comparison of the Dutch, French and Latin texts clearly shows.
How can you tell if someone has used this French translation? One approach is to look for words that are specific for the French text. Huygens writes in his letter to Van Schooten 'Doch sy luyden' when talking about the French mathematicians, Van Schooten translates this correctly as 'Cæterum illi'. Korteweg however translates 'Mais ces savants' which adds an extra explanation to the translation. David has 'But these savants' and Dupont & Roero 'Tuttavia questi illustri studiosi'. And there are other cues that give away on which source a translation is based.
The French translation by Korteweg of the Dutch text was probably intended to convey the meaning of the Dutch text, and not so much to give an idea of the phrasing. See e.g. the first three propositions where Ick ... hebbe had to give way to the rather impersonal Avoir. Apart from an occasional spelling error (see proposition IX), there is a sometimes less than happy choice of wording (eg proposition XIV: where one reads rapport for reden, in stead of ratio, raison or proportion).
Sometimes there is also some (unnecessary) explanation involved. Propositions XIII and XIV both started originally with Als. In the French translation of proposition XIV this becomes Si as one would expect. Proposition XIII however starts rather unexpectedly with Dans l'hypothèse que. I rest my case.

F.N. David, 1962; p 116-119

It was the translation of FN David that turned out to be the greatest surprise though not a happy one. It is suggested on pages 113 and 115 of her book that it is a translation from the Latin text of Van Schooten. It now appears however to be a translation of the French translation by Korteweg. This puts the remark of Anders Hald (1990, p 69) about the differences in wording between his translation from the Dutch text and Davids translation in quite another perspective.
In the translation of proposition IX, the last sentence is left out for no appearent reason.
In the translation of problem III, the last sentence is probably left out because David suggests in a footnote on p 119 that both Huygens and Todhunter are wrong in their calculation of the answer.

A. Hald, 1990, p 69-78

Halds translation albeit not complete, is reliable in meaning and stays as close as possible to the way Huygens phrased his idea's in Dutch. In problem V Anders translates the number of chances in stead of the chance.

Huygens' propositions

1. 1. Als ick gelijcke kans hebbe om a of b te hebben, dit is my so veel weerdt als .
   2. Si a vel b expectem, quorum utrumvis æquè facilè mihi obtingere possit, expectatio mea dicenda est (a+b)/2.
   3. If I expect a or b, either of which, with equal probability, may fall to me, then my Expectation is worth that is, the half Sum of a and b.
   4. Si a vel b expectem, quorum utrumvis æquè facilè mihi obtingere possit, expectatio mea dicenda est (a+b)/2.
   5. If I expect a or b, and have an equal Chance of gaining either of them, my Expectation is worth .
   6. Avoir des chances égales d'obtenir a ou b me vaut .
   7. To have equal chances of getting a and b is worth (a+b)/2.
   8. If I have equal chances of getting a or b, this is so much worth to me as (a+b)/2.
2. 1. Als ick gelijcke kans hebbe tot a of b of c, het is my soo veel weerdt als of ick hadde.
   2. Si a, b vel c expectem, quorum unumquodque pari facilitate mihi obtingere possit, expectatio mea æstimanda est (a+b+c)/3.
   3. If I expect a, b, or c, either of which, with equal facility, may happen, then the Value of my Expectation is worth or the third part of the Sum of a b and c.
   4. Si a, b vel c expectem, quorum unumquodque pari facilitate mihi obtingere possit, expectatio mea æstimanda est (a+b+c)/3.
   5. If I expect a, b, or c, and each of them be equally likely to fall to my Share, my Expectation is .
   6. Avoir des chances égales d'obtenir a, b ou c me vaut .
   7. To have equal chances of getting a, b and c is worth (a+b+c)/3.
   8. If I have equal chances of getting a, b or c, this is so much worth to me as if I had (a+b+c)/3.
3. 1. Als het getal der kanssen die ick hebbe tot a is p, ende het getal der kanssen die ick tot b heb is q, nemende altijdt dat ieder kans even licht kan gebeuren; Het is my weerdt .
   2. Si numerus casuum, quibus mihi eveniet a, sit p; numerus autem casuum, quibus mihi eveniet b, sit q, sumendo omnes casus æquè in proclivi esse; expectatio mea valebit (pa+pb)/(p+q). p should be q
   3. If the Number of Chances by which a falls to me, be p, and the Number of Chances by which b, be q, and supposing all the Chances do happen with equal facility, then the Value of my Expectation is , i.e. the Product of a multiplied in the Number of its Chances added to the Product of b multiplied into the Number of its Chances and the Summ divided by the number of Chances both of a and b.
   4. Si numerus casuum, quibus mihi eveniet a, sit p; numerus autem casuum, quibus mihi eveniet b, sit q, sumendo omnes casus æquè in proclivi esse; expectatio mea valebit (pa+qb)/(p+q).
   5. If the number of Chances I have to gain a, be p, and the number of Number of Chances both of a and b. Chances I have to gain b, be q. Supposing the Chances equal; my Expectation will then be worth .
   6. Avoir p chances d'obtenir a et q d'obtenir b, les chances étant égales, me vaut .
   7. To have p chances of obtaining a and q chances of obtaining b, chances being equal, is worth (pa+qb)/(p+q).
   8. If the number of chances of getting a is p and the number of chances of getting b is q, assuming always that any chance occurs equally easy, then this is worth (pa+qb)/(p+q) to me.
4. 1. Genomen dan dat ick tegens een ander speele ten dryen uyt en dat ick alreede 2 spelen hebbe en hy maer een. Ick wil weeten, ingevalle wy het spel niet en wilden voortspeelen, maer hetgeen ingeset is gerechtelijck wilden deelen, hoeveel my daer van komen soude.
   2. Ut igitur ad primò propositam quæstionem veniamus, nimirum, de facienda distributione inter diversos collusores, quando eorum sortes inæquales sunt, opùs est ut à facilioribus incipiamus.
      NB the actual text should be:
      Sumpto itaque me cum aliquo certare, hoc pacto: ut qui prius ter vicerit, quod depositum est, lucretur, & me jam bis vicisse, alterum vero semel. Scire cupio, si lusum prosequi non velimus, sed pecuniam, de qui certamus, prout æquum est, partiri, quantum ejus mihi obtingeret.
   3. That I may come to the Question propos'd, viz. The making a just Distribution amongst Gamesters, when their Hazards are unequal; We must begin with the most easy Cases.
      NB the actual text should be:
      SUppose then I play with another, on condition that he who wins the three first Games shall have the Stakes, and that I have gain'd two and he one, I would know, if we agree to break off the Game and part the Stakes justly, how much falls to my Share?
   4. Sumpto itaque me cum aliquo certare, hoc pacto: ut qui prius ter vicerit, quod depositum est, lucretur, & me jam bis vicisse, alterum vero semel. Scire cupio, si lusum prosequi non velimus, sed pecuniam, de qui certamus, prout æquum est, partiri, quantum ejus mihi obtingeret.
   5. To come to the Question first propos'd, How to make a fair Distribution of the Stake among the several Gamesters, whose Chances are unequal? The best way will be to begin with the most easy Cases of that Kind.
      NB the actual text should be:
      SUPPOSING therefore that I play with another upon this Condition, That he who gets the first three Games shall have the Stake; and that I have won two of the three, and he only one. I desire to know, if we agree to leave off and divide the Stake, how much falls to my Share?
   6. Supposons que je joue contre une autre personne à qui aura gagné le premier trois parties, et que j'aie déja gagné deux parties et lui une. Je veux savoir quelle partie de l'enjeu m'est due en cas où nous voulons interrompre le jeu et partager équitablement les mises.
   7. Suppose I play against an opponent as to who will win the first three games and that I have already won two and he one. I want to know what proportion of the stakes is due to me if we decide not to play the remaining games.
   8. Suppose that I play against another person about three games, and that I have already won two games and he one. I want to know what my proportion of the stakes should be, in case we decide not to continue the play and divide the stakes equitably between us.
5. 1. Zij gestelt dat my 1 spel ontbreeckt, en die tegens my speelt 3 spelen. Nu moet men de verdeeling maecken.
   2. Ponamus unum mihi deficere ludum & collusori meo tres lusus. Oportet hîc facere distributionem.
   3. Suppose I want but one Game, and my Fellow-gamester three, it is required to make a just Distribution of the Stake:
   4. Ponamus unum mihi deficere ludum & collusori meo tres lusûs. Oportet hïc facere distributionem.
   5. Suppose I want one Game of being up, and my Adversary wants three; How must the Stake be divided?
   6. Supposons qu'il me manque une partie à moi et trois à mon adversaire. Il s'agit de partager l'enjeu dans cette hypothèse.
   7. Suppose that I lack one point and my opponent three. what proportion of the stakes, etc.
   8. ---
6. 1. Zij gestelt dat my twee spelen ontbreecken, en hem die tegen my speelt drie spelen.
   2. Ponamus mihi deficere duos lusus & collusori meo tres lusus.
   3. Suppose I want two Games, and my Fellow-gamester three.
   4. Ponamus mihi deficere duos lusûs & collusori meo tres lusûs.
   5. Suppose I have two Games to get, and my Adversary three.
   6. Supposons qu'il me manque deux parties et qu'il en manque trois à mon adversaire.
   7. Suppose that I lack two points and my opponent three, etc.
   8. ---
7. 1. Zij gestelt dat aen my noch twee spelen ontbreecken, en hem 4 spelen.
   2. Ponamus mihi deficere duos lusus & collusori meo quattuor.
   3. Let us suppose I want two Games, and my Adversary four.
   4. Ponamus mihi deficere duos lusûs & collusori meo quattuor.
   5. Suppose I want two Games, and my Fellow four.
   6. Supposons qu'il me manque encore deux parties et lui quatre.
   7. Suppose that I lack two points and my opponent four, etc.
   8. ---
8. 1. Laet ons nu stellen dat drie persoonen t'samen speelen, daer van den eersten 1 spel ontbreeckt, den tweeden mede 1 spel, maer den derden 2 spelen.
   2. Nunc vero ponamus tres esse collusores, quorum primo ut & secundo unus lusus deficiat, sed tertio duos lusus.
   3. Let us suppose three Gamesters whereof the first and second want 1 Game, but the third 2.
   4. Nunc vero ponamus tres esse collusores, quorum primo ut & secundo unus lusus deficiat, sed tertio duos lusûs.
   5. Suppose now there are three Gamesters and that the first and second want a Game a piece, and the third wants two Games.
   6. Supposons maintenant que trois personnes jouent ensemble et qu'il manque une partie à la première ainsi qu'à la deuxième mais qu'il en manque deux à la troisième.
   7. Suppose now that three people play and that the first and the second lack one point each and the third two points.
   8. Let us suppose that three people play together, and that the first lacks one game and the second one game and the third two games.
9. 1. Om tusschen soo veel speelders als voor-gestelt zijn, waer van d'eene meer en d'ander minder speelen ontbreecken een ieder haer deel te vinden, soo moet ingesien worden, wat hem, wiens deel men begeert te weeten, soude toekomen, indien of hy, of elck van d'andere in 't besonder het eerste volgende spel quam te winnen. Dit dan alles saemen geaddeert en door het getal der speelders gedeelt, soo komt het gesochte gedeelte van den eenen.
   2. Ut tot collusores, quot quis voluerit, ex quibus uni plures & alii pauciores lusus deficiunt, cujusque pars inveniatur, considerandum est, quid illi, cujus partem invenire volumus, deberetur, si vel ipse, vel quilibet, reliquorum primum sequentem ludum vinceret. Hæ autem partes si in unam summam colligantur, & aggregatum per numerum collusorum dividatur, quotiens ostendit unius quæsitam partem.
   3. In any Number of Gamesters you please, amongst whom there are some who want more, some fewer Games: To find what is any ones Share in the Stake, we must consider what would be due to him, whose Share we investigate, , if either he, or any of his Fellow-Gamesters should gain the next following Game; add all their Shares together, and divide the Sum by the Number of Gamesters, the Quotient is his Share you were seeking.
   4. Ut tot collusores, quot quis voluerit, ex quibus uni plures & alii pauciores lusûs deficiunt, cujusque pars inveniatur, considerandum est, quid illi, cujus partem invenire volumus, deberetur, si vel ipse, vel quilibet, reliquorum primum sequentem ludum vinceret. Hæ autem partes si in unam summam colligantur, & aggregatum per numerum collusorum dividatur, quotiens ostendit unius quæsitam partem.
   5. To find the several Shares of as many Gamesters, as we please, some of which shall want more Games, others fewer; we must consider what he, whose Share we want to find, wou'd gain, if he, or any one of the others wins the next Game: Then adding together what he wou'd gain in all those particular Cases, and dividing the Sum by the Number of Gamesters, the Quotient gives the particular Share required.
   6. Pour calculer la part de chacun d'un nombre donné de (sic!) joueurs, auxquels manquent de parties en nombres donnés pour chacun d'eux séparément, il faut d'abord se rendre compte de ce qui reviendrait à celui dont on veut savoir la part dans le cas où lui et dans ceux où chacun des autres à son tour aurait gagné la première partie suivante. En ajoutant toutes ces parts et en divisant la somme par le nombre des joueurs on trouve la part cherchée du joueur considéré
   7. In order to calculate the proportion of the stakes due to each of a given number of players who are each given a number of points short, it is necessary, to begin with, to consider what is owing to each in turn in the case where each might have won the succeeding game. At this point David omits the last part of this Proposition in her translation.
   8. To calculate the proportion due to each of a given number of players, who each lacks a given number of games, it is necessary to find out what the player, whose proportion we want to know, would get if he or any of the other players wins the following game. Adding these parts and dividing by the number of players give us the proportion sought.
10. 1. Te vinden van hoeveel reysen men kan neemen een 6 te werpen met eene steen.
    2. Invenire, quot vicibus suscipere quis possit, ut unâ tesserâ 6 puncta jaciat.
    3. To find how many times one may undertake to throw 6 with One Dye.
    4. Invenire, quot vicibus suscipere quis possit, ut unâ tesserâ 6 puncta jaciat.
    5. To find how many Throws one may undertake to throw the Number 6 with a single Die.
    6. Trouver en combien de fois l'on peut accepter de jeter un six avec un dé.
    7. To find how many times one may wager to throw a six with one die.
    8. To find how many turns one should take to throw a six with one die.
11. 1. Te vinden van hoeveel reysen men kan neemen 2 sessen te werpen met 2 steenen.
    2. Invenire, quot vicibus suscipere quis possit, ut duabus tesseris 12 puncta jaciat.
    3. To find how many times one may undertake to throw 12 with Two Dice.
    4. Invenire, quot vicibus suscipere quis possit, ut duabus tesseris 12 puncta jaciat.
    5. To find in how many Throws one may venture to throw the Number 12 with two Dice.
    6. Trouver en combien de fois l'on peut accepter de jeter 2 six avec 2 dés.
    7. To find how many times one should wager to throw 2 sixes with 2 dice.
    8. To find how many turns one should take to throw two sixes with 2 dice.
12. 1. Te vinden met hoe veel steenen men kan nemen ten eersten 2 sessen te werpen.
    2. Invenire, quot tesseris suscipere quis possit, ut primâ vice duos senarios jaciat.
    3. To find with how many Dice one can undertake to throw two Sixes at the first Cast.
    4. Invenire, quot tesseris suscipere quis possit, ut prima vice duos senarios jaciat.
    5. To find with how many Dice one may undertake to throw two Sixes the first Throw.
    6. Trouver le nombre de dés avec lequel on peut accepter de jeter 2 six du premier coup.
    7. To find the number of dice with which one may wager to throw 2 sixes at the first throw.
    8. To find how many dice one should take to throw two sixes at the first throw.
13. 1. Als ick tegen een ander speel met 2 steenen alleen eene werp, op conditie, dat, indien der 7 oogen komen, ick winnen sal; maer hy indiender 10 oogen komen; en ingevalle iets anders, dat wy dan gelijckelijck deelen sullen hetgeen ingeset is: Te vinden wat deel daer van ons elck toekomt.
    2. Si cum alio ludam duabus tesseris unum solummodo jactum, hâc conditione, ut si septenarius eveniat, ego vincam; at ille, si denarius obtingat; si verò quidquam aliud accidat, ut tum id quod depositum est æqualiter dividamus: Invenire qualis istius pars cuique nostrum debeatur.
    3. If I am to play with another One Throw, on this condition, that if 7 comes up I gain, if 10 he gains; if it happens that we must divide the Stake, and not play, to find how much belongs to me, and how much to him.
    4. Si cum alio ludam duabus tesseris unum solummodò jactum, hâc conditione, ut si septenarius eveniat, ego vincam; at ille, si denarius obtingat; si verò quidquam aliud accidat, ut tum id quod depositum est æqualiter dividamus: Invenire qualis istius pars cuique nostrûm debeatur.
    5. Supposing I lay with another to take one Throw with a pair of Dice upon these Terms, That if the Number 7 comes up, I shall win, and if 10 comes up, he shall win; and after this Bargain made, we consent to draw Stakes by a fair Division, according to the Value of our Chances in the present Contract: To find what shall be our several Shares.
    6. Dans l'hypothèse que je joue un coup de deux dés contre une autre personne à condition, que s'il vient 7 points, j'aurai gagné, mais qu'elle aura gagné s'il en vient 10, et que nous partagerons l'enjeu en parties égales s'il vient autre chose, trouver la part qui revient à chacun de nous.
    7. On the hypothesis that I play a throw of 2 dice against an opponent with the rule that if the sum is 7 points I will have won but that if the sum is 10 he will have won, and that we split the stakes in equal parts if there is any other sum, find the expectation of each of us.
    8. Suppose that I play one single throw with two dice against another person on the condition that if the outcome is 7 points I win, if the outcome is 10 points he wins, and otherwise the stakes should be divided equally between us. To find what proportion each of us should have.
14. 1. Als ick en noch een ander met beurten werpen met 2 steenen, ende bespreecken, dat ick sal winnen, soo haest ick 7 ooghen werp, ende hy, soo haset, als hy 6 ooghen werpt, mits dat ick hem de voorwerp geve. Te vinden in wat reden mijn kans tegen de sijne staet.
    2. Si ego & alius duabus tesseris alternatim jaciamus, hâc conditione, ut ego vincam simul atque septenarium jaciam, ille verò quàm primum senarium jaciat; ita videlicet, ut ipsi primum jactum condedam: Invenire rationem meæ ad ipsius sortem.
    3. If I were playing with another by turns with two Dice, on this condition, that if I throw 7 I gain, and if he throw (sic!) 6 he gains, allowing him the first Throw: To find the proportion of my Hazard to his.
    4. Si ego & alius duabus tesseris alternatim jaciamus, hac conditione, ut ego vincam simul atque septenarium jaciam, ille verò quâm primûm senarium jaciat; ita videlicet, ut ipsi primum jactum condedam: Invenire rationem meæ ad ipsius sortem.
    5. If my self and another play by turns with a pair of Dice upon these Terms, That I shall win if I throw the Number 7, or he if he throws 6 soonest, and he to have the Advantage of the first Throw: To find the Proportion of our Chances.
    6. Si un autre joueur et moi jettent tour à tour 2 dés à condition que j'aurai gagné dès que j'aurai jeté 7 points et lui dès qu'il en aura jeté 6, tandis que je lui laisse le premier coup, trouver le rapport de ma chance à la sienne.
    7. If another player and I throw turn and turn about with 2 dice on condition that I will have won when I have thrown 7 points and he will have won when he has thrown 6, if I let him throw first find the ratio of my chance to his.
    8. Suppose that I and another player take turns in throwing with 2 dice on condition that I win if I throw 7 points and he wins if he throws 6 points, and I let him have the first throw. To find the ratio of my chance to his.

Huygens' problems for the reader

1. 1. A en B speelen teghen malkander met 2 steenen, op dese conditie: dat A sal winnen als hy 6 oogen werpt, maer B sal winnen als hy 7 oogen werpt. A sal eerst eene werp doen; daernae B twee werpen achtervolgens; dan weder A 2 werpen; en soo voorts, tot dat d'een of d'ander sal winnen. De vrage is in wat reden de kans van A staet tegen die van B? antw. als 10355 tot 12276.
   2. A & B unà ludunt duabus tesseris, hâc conditione, ut A vincat, si senarium jaciat, at B si septenarium jaciat. A primò unum jactum instituat; deinde B duos jactus consequenter; tum rursus A duos jactus, atque sic deinceps, donec hic vel ille victor evadat. Quæritur ratio sortis ipsius A ad sortem ipsius B? Resp. ut 10355 ad 12276.
   3. A and B play together with two Dice, A wins if he throws 6, and B if he throws 7; A at first gets one Throw, then B two, then A two , and so on by turns, till one of them wins. I require the proportion of A's Hazard to B's? Answer, It is as 10355 to 12276.
   4. A & B unà ludunt duabus tesseris, hâc conditione, ut A vincat, si senarium jaciat, at B si septenarium jaciat. A primò unum jactum instituat; deinde B duos jactus consequenter ; tum rursùs A duos jactus, atque sic deinceps, donec hic vel ille victor evadat. Quaeritur ratio sortis ipsius A ad sortem ipsius B? Resp. ut 10 355 ad 12 276.
   5. A and B play together with a pair of Dice upon this Condition, That A shall win if he throws 6, and B if he throws 7; and A is to take one Throw first, and then B two Throws together, then A to take two Throws together, and so on both of them the same, till one wins. The Question is, What Proportion their Chances bear to one another? Answ. As 10355 to 12276.
   6. A et B jouent ensemble avec 2 dés à la condition suivante: A aura gagné s'il jette 6 points, B s'il jette en 7. A sera le premier un seul coup; ensuite B 2 coups successifs; puis de nouveau A 2 coups, et ainsi de suite, jusqu'à ce que l'un ou l'autre aura gagné On demande le rapport de la chance de A à celle de B? Response: comme 10355 est à 12276
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   8. A and B play against each other with two dice on the condition that A wins if he throws six points and B wins if he throws seven points. A has the first throw. B the following two, then A the following two, and so on, until one or the other wins. The Question is, What is the ratio of A's chances to B's? Answer: 10,355 to 12,276.
2. 1. Drie speelders A, B en C nemende 12 schijven, van welcke 4 wit zijn en 8 swart, speelen op conditie, dat die van haer blindelings eerst een witte schyve sal gekosen hebben winnen sal, en dat A de eerste sal nemen, B de tweede, en dan C, en dan wederom A, en soo vervolgens met beurten. De vraghe is in wat reden haere kanssen staen tegens malkander?
   2. Tres collusores A, B & C assumentes 12 calculos, quorum 4 albi & 8 nigri existunt, ludunt hac conditione: ut, qui primus ipsorum velatis oculis album calculum elegerit, vincat; & ut prima electio sit penes A, secunda penes B & tertia penes C, & tum sequens rursus penes A, atque sic deinceps alternatim. Quaeritur, quaenam futura sit ratio illorum sortium
   3. Three Gamesters, A, B, and C, take 12 Counters, of which are four white and eight black; the Law of this Game is this, that he shall win, who, hood-wink'd, shall first chuse a white Counter, and that A shall have the first Choice, B the second, and C the third; and so, by turns, till one of them win (sic!). Quær.What is the Proportion of their Hazards?
   4. Tres Collusores A, B & C assumentes 12 calculos, quorum 4 albi & 8 nigri existunt, ludunt hac conditione: & ut, qui primus ipsorum velatis oculis album calculum elegerit, vincat; ut prima electio sit penès A, secunda penès B & tertia penès C, & tum sequens rursus penes A, atque sic deinceps alternatim. Quaeritur, quaenam futura sit ratio illorum sortium
   5. THREE Gamesters, A, B, and C, taking 12 Counters, 4 of which are white, and 8 black, play upon these Terms: That the first of them that shall blindfold choose a white Counter shall win; and A shall have the first Choice, B the second, and C the third; and then A to begin again, and so on in their turns. What is the Proportion of their Chances?
   6. Trois Joueurs A, B et C prennent 12 jetons dont 4 blancs et 8 noirs; il jouent à cette condition que celui gagnera qui aura le premier, en choisissant à l'aveuglette, tiré un jeton blanc, et que A choisira le premier, B ensuite, puis C, puis de nouveau A et, ainsi de suite, à tour de rôle. On demande le rapport de leur chances?
   7. Three gamblers A, B, and C take 12 balls of which 4 are white and 8 black. They play with the rules that the drawer is blindfold, A is to draw first, then B and then C, the winner to be the one who first draws a white ball. What is the ratio of their chances?
   8. Three players A, B and C, having 12 chips of which four are white and eight black, play on the condition that the first blindfolded player to draw a white chip wins, and that A draws first, B next, and then C, then A again, and so on. The Question is, What are the ratios of their chances to each other?
3. 1. A wed tegens B, dat hy uyt 40 kaerten, dat is, 10 van ieder soort, 4 kaerten uyttrecken sal, soo dat hy van elcke soorte een sal hebben. Hier wordt de kans van A tegen B gevonden, als 1000 tegen 8139.
   2. A certat cum B quòd ipse ex 40 chartis lusoriis, id est, 10 cujusque specei, 4 chartas extracturus sit; ita ut ex unaquaque specie habeat unam. Et invenitur ratio sortis A ad sortem B ut 1000 ad 8139.
   3. A wagers with B, that of 40 cards, that is, 10 of every Suit, he will pick out four, so that there shall be one of every Suit; A's Hazard to B's, in this Case, is as 1000 to 8139.
   4. A certat cum B quòd ipse ex 40 chartis lusoriis, id est, 10 cujusque specei, 4 chartas extracturus sit; ita ut ex unaquaque specie habeat unam. Et invenitur ratio sortis A ad sortem B ut 1000 ad 8139.
   5. A lays with B, that out of 40 cards, i.e. 10 of each different Sort, he will draw 4, so as to have one of every Sort. And the Proportion of his Chance to that of B, is found to be as 1000 to 8139.
   6. A parie contre B, que de 40 cartes, dont dix de chaque couleur, il en tirera 4 de manière à en avoir une de chaque couleur. On trouve dans ce cas que le chance de A est à celle de B comme 1000 est à 8139.
   7. A wagers B that, given 40 cards of which 10 are of one colour, 10 of another, 10 of another and 10 of another, he will draw 4 so as to have one of each colour. At this point David omits the last part of this Problem in her translation.
   8. A wagers with B that out of 40 cards, there being 10 of each colour, he will draw four cards so that he gets one of each colour. The chances of A to those of B are found to be 1000 to 8139.
4. 1. Genomen hebbende ghelijck hier te vooren 12 schyven, 4 witte en 8 swarte; soo wed A tegen B dat hij blindelings 7 schyven sal daer uyt nemen, onder welcke 3 witte sullen sijn. Men vraegt in wat reden de kans van A staet tegen die van B.
   2. Assumptis, ut ante, 12 calculis, 4 albis & 8 nigris, certat A cum B, quòd velatis oculis 7 calculos ex iis exempturus sit, inter quos 3 albi erunt. Quæritur ratio sortis ipsius A ad sortem ipsius B.
   3. Supposing as before, 4 white Counters and 8 black, A wagers with B that out of them, he shall pick 7 Counters, of which there are 3 white. I require the proportion of A's Hazard to B's?
   4. Assumptis, ut ante, 12 calculis, 4 albis & 8 nigris, certat A cum B, quod velatis oculis 7 calculos ex iis exempturus sit, inter quos 3 albi erunt. Quæritur ratio sortis ipsius A ad sortem ipsius B.
   5. HAVING chosen 12 Counters as before, 8 black and 4 white, A lays with B that he will blindfold take 7 out of them, among which there shall be 3 black (sic!) ones. Quaere, What is the Proportion of their Chances?
   6. On prend comme plus haut 12 jetons dont 4 blancs et 8 noirs. A parie contre B que parmi 7 jetons qu'il en tirera à l'aveuglette, il se trouvera 3 blancs. On demande le rapport de la chance de A à celle de B.
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   8. As before, the players have 12 chips, four are white and eight black; A wagers with B that by drawing seven chips blindfolded, he will get three white chips. The Question is, What is the ratio of A's chances to B's?
5. 1. A en B genomen hebbende elck 12 penningen spelen met 3 dobbelsteenen op dese conditie: dat als'er 11 oogen geworpen worden, A een penning aen B moet geven; maer als'er 14 geworpen worden, dat dan B een penning aen A moet geven; en dat hy het spel winnen sal, die eerst al de penningen sal hebben. Hier werdt ghevonden de kans van A tegen die van B te zijn, als 244140625 tot 282429536481.
   2. A & B assumentes singuli 12 nummos ludunt tribus tesseris, hâc conditione: ut, si 11 puncta jaciantur, A tradat nummum ipsi B; at si 14 puncta jaciantur B tradat nummum ipsi A; & ut ille ludum victurus sit, qui primùm omnes habuerit nummos. Et invenitur ratio sortis ipsius A ad sortem ipsius B, ut 244140625 ad 282429536481.
   3. A and B taking 12 Counters, play with 3 Dice after this manner, that if 12 (sic!) comes up, A shall give one Counter to B, but if 14 comes up, then B shall give one to A; and that he shall gain who first has all the Counters. A's Hazard to B's is 244140625 to 282429536481.
   4. A & B assumentes singuli 12 nummos ludunt tribus tesseris, hac conditione: ut, si 11 puncta jaciantur, A tradat nummum ipsi B; at si 14 puncta jaciantur B tradat nummum ipsi A; & ut ille ludum victurus sit, qui primùm omnes habuerit nummos. Et invenitur ratio sortis ipsius A ad sortem ipsius B, ut 244140625 ad 282429536481.
   5. A and B taking 12 Pieces of Money each, play with 3 Dice on this Condition, That if the Number 11 is thrown, A shall give B one Piece, but if 14 be thrown, then B shall give one to A; and he shall win the Game that first gets all the Pieces of Money. And the Proportion of A's Chance to B's is found to be, as 244,140,625 to 282,429,536,481.
   6. Ayant pris chacun 12 jetons, A et B jouent avec 3 dés à cette condition qu'à chaque coup de 11 points, A doit donner un jeton à B, mais que B en doit donner 1 à A à chaque coup de 14 points, et celui là gagnera qui sera premier en possession de tous les jetons. On trouve dans ce cas que le chance de A est à celle de B comme 244140625 est à 282429536481.
   7. David gives only a paraphrase"... A and B start with 12 balls and continue to throw three dice on the condition that if 11 is thrown A gives a ball to B and if 14 is thrown B gives one to A. The game is to continue until one or the other has all the balls." Again David omits the last sentence which contains the answer
   8. A and B having 12 counters play with three dice on the condition, that if 11 points are thrown, A gives a counter to B and if 14 points are thrown, B gives a counter to A and that he wins the play who first has all the counters. Here it is found that the number of chances of A to that of B is 244,140,625 to 282,429,536,481.

References

• 1656-1657,1660
  • Huygens, Chr (1655-1666, 1920) Oeuvres Complètes de Christiaan Huygens, tome 14. La Haye, Martinus Nijhoff.
  • Huygens, Chr (1660, 1998) Van Rekeningh in Spelen van Geluck vertaald en toegelicht door Wim Kleijne. Epsilon Uitgaven, Utrecht.
• 1657
  • Waerden, BL van der (ed, 1975) Die Werke von Jacob Bernoulli, Band 3, Birckhauser Verlag Basel
  • Schooten, Frans van. (1657) Exercitationum mathematicarum libri quinque. Leiden: J. Elsevir,
  • The first page of De ratiociniis in ludo aleae (1657).
• 1692
  • Arbuthnot, J (1692) Of the Laws of Chance or a Method of Calculation of the Hazard of Game,.... Londen: B. Motte.
• 1713
  • Waerden, BL van der (ed, 1975) Die Werke von Jacob Bernoulli, Band 3, Birckhauser Verlag Basel
• 1714
  • Posner library: 519 H98L 1714 "... Libellus de ratiociniis in ludo aleae; or The value of all chances in games of fortune; cards, dice, wagers, lotteries, &c., mathematically demonstrated."
  • Libellus de Ratiociniis in Ludo Aleae (pdf file)
• 1920
  • Huygens, Chr (1655-1666, 1920) Oeuvres Complètes de Christiaan Huygens, tome 14. La Haye, Martinus Nijhoff.
• 1962
  • David, FN (1962, 1998) Gods, Games and Gambling: a History of Probability and Statistical Ideas. Mineola, N.Y., Dover Publications, inc
• 1990
  • Hald, A (1990) A History of Probability and Statistics and their Applications before 1750 New York, Wiley