324 SAMUEL CLARK. 



omitted, and many examples and illustrations are given in order 

 to render the subject accessible to persons not very far advanced 

 in mathematics. 



The book presents nothing that is new and important. The 

 game of bowls seems to have been a favourite with Clark ; he 

 devotes his pages 44 — 68 to problems connected with this game. 

 He discusses at great length the problem of finding the chance of 

 throwing an assigned number of points with a given number of 

 similar dice; see his pages 113 — ISO. He follows Simpson, but 

 he also indicates De Moivre's Method ; see Art. 364. Clark 

 begins the discussion thus : 



In order to facilitate the solution of this and the following problem, 

 I shall lay down a lemma which was communicated to me by my inge- 

 nious friend Mr William Fayne, teacher of mathematics. 



The Le7tima. 



The sum of 1, 3, 6, 10, 15, 21, 28, 36, &c. continued to (n) number 



J, ^ . , ^ n + 2 n + 1 n 



01 terms is equal to — ^ — x — ^— x - . 



It was quite unnecessary to appeal to William Pa3riie for such 

 a well-known result ; and in fact Clark himself had given on his 

 page 84 Newton's general theorem for the summation of series ; 

 see Art. 152. 



Clark discusses in his pages 139 — 153 the problem respecting 

 a run of events, which we have noticed in Art. 325. Clark detects 

 the slight mistake which occurs in De Moivre's solution ; and from 

 the elaborate manner in which he notices the mistake we may 

 conclude that it gave him great trouble. 



Clark is not so fortunate in another case in which he ventures 

 to differ with De Moivre ; Clark discusses De Moivre's Problem ix. 

 and arrives at a different result ; see Art. 269. The error is 

 Clark's. Taking De Moivre's notation Clark assumes that A must 

 either receive q G from B, or pay jiL to B. This is wrong. Sup- 

 pose that on the whole A wins in 5' -F m trials and loses in m trials ; 

 then there is the required difference of q games in his favour. In 

 this case he receives from B the sum {(i + 111) G and pays to him 

 the sum m.L ; thus the balance is qG + m {G — L) and not qG SiS 

 Clark says. 



