52 ARBUTHNOT. 



and he arrives at the same result as James Bernoulli on page 58 

 of the Ars Conjectandi. Our author adds, 



I have suppos'd here the Sense of the Problem to be, that when any- 

 one chus'd a Counter, he did not diminish their number; but if he 

 miss'd of a white one, put it in again, and left an equal hazard to him 



who had the following choice; for if it be otherwise suppos'd, ^'s share 



55 9 



will be Y9^ » which is less than Yq • 



55 



This result ^-^ however is wrong in either of the other two 



senses which James Bernoulli ascribes to the problem, for which he 



77 101 

 obtains j^ and z-^ respectively as the results ; see Art. 35. 



84. Then follow some other calculations about games. We 

 have some remarks about the Boyal-Oak Lottery which are analo- 

 gous to those made on the Play of the Royal Oak by De Moivre 

 in the Preface to his Doctrine of Chances. 



A table is g^iven of the number of various throws which can be 

 made with three dice. Pages 84 — 39 are taken from Pascal ; they 

 seem introduced abruptly, and they give very little that had not 

 already occurred in the translation of Huygens's treatise. 



85. Our author touches on Whist ; and he solves two problems 

 about the situation of honours. These solutions are only approxi- 

 mate, as he does not distinguish between the dealers and their 

 adversaries. And he also solves the problem of comparing the 

 chances of two sides, one of which is at eight and the other at 

 nine; the same remark applies to this solution. He makes the 

 chances as 9 to 7; De Moivre by a stricter investigation makes 

 them nearly as 25 to 18. See Doctrine of Chances, page 176. 



86. Our author says on page 43, 



All the former Cases can be calculated by the Theorems laid down 

 by Monsieur Huygens; but Cases more compos'd require other Prin- 

 ciples; for the easy and ready Computation of which, I shall add one 

 Theorem more, demonstrated after Monsieur Huygens's method. 



The theorem is : " if I have p Cliances for a, q Chances for h, 



