SIMPSON. 211 



870. Simpson's Problem xxx. is as follows : 



Two Gamesters, A and £, equally skilful, enter into Play together, 

 and agree to continue the same till (n) Games are won and lost. 'Tis 

 required to find the Probability that neither comes off a Winner of 

 Q'Jn Stakes, and also the Probability that B is never a Winner of 

 that Number of Stakes during the whole Time of the Play; r being 

 a given, and n any very great, Number, 



Simpson says in bis Preface relating to bis Problems XXIV. and 

 XXX. tbat tbey 



" are the same with the two new ones, added in the End of Mr 

 Be Moivre's last Edition, whose Demonstrations that learned Author 

 was pleased to reserve to himself, and are here fully and clearly in- 

 vestigated...." 



The same two problems are thus referred to in Simpson's 

 title page : 



Full and clear Investigations of two Problems, added at the end of 

 Mr. De Moivre's last Edition ; one of them allowed by that great Man 

 to be the most useful on the Subject, but their Demonstrations there 

 omitted. 



Simpson is quite wrong in claiming tbe solution of Pro- 

 blem XXX, and saying that De Moivre had reserved his demon- 

 stration to himself. The investigation is that for determining the 

 approximate value of terms near the largest in the expansion of 

 {a + hy ; it is given in the Doctrine of Chances, second edition, 

 pages 233 — 243, third edition pages 241 — 2ol : the method of 

 Simpson is in fact identical with De Moivre's. 



871. We may remark that Simpson published a work in l7o7 

 under the title of Miscellaneous Tracts on some curious, and 

 very interesting Subjects in Mechanics, Physical- Astronomy, and 

 Speculative Mathematics ; ... 



In this work on pages 64 — 75 we have a section entitled An 

 Attempt to shew the Advantage arising by Taking the Mean of a 

 Number of Observations, in Practical Astronomy. 



This is a very interesting section ; the problems solved by 

 Simpson were reproduced by Lagrange in a memoir in the fifth 

 volume of the Miscellanea Taurinensia, without any allusion how- 

 ever to Simpson. 



14—2 



