MONTMORT. 91 



it begins with a fraction having twelve figures in its denominator, 

 which in its simplest form would only have four figures. 



In the only copy of the first edition which I have seen these 

 columns are given correctly ; in both editions the description given 

 in the text corresponds not to the incorrect forms but to the cor- 

 rect forms. 



159. Montmort next discusses the game of Lansquenet; this 

 discussion occupies pages 105 — 129. It does not appear to present 

 any point of interest, and it would be useless labour to verify the 

 complex arithmetical calculations which it involves. A few lines 

 which occurred on pages 40 and 41 of Montmort's first edition are 

 omitted in the second ; while the Articles 84 and 95 of the second 

 edition are new. Ai'ticle 84 seems to have been suggested to 

 Montmort by John Bernoulli ; see Montmort's page 288 : it relates 

 to a point which James Bernoulli had found difficult, as we have 

 already stated in Art. 119. 



160. Montmort next discusses the game of Treize ; this dis- 

 cussion occupies pages 130 — 143. The problem involved is one of 

 considerable interest, which has maintained a permanent place in 

 works on the Theory of Probability. 



The following is the problem considered by Montmort. 



Suppose that we have thirteen cards numbered 1, 2, 3 ... up to 

 13 ; and that these cards are thrown promiscuousl}^ into a bag. 

 The cards are then drawn out singly ; required the chance that, 

 once at least, the number on a card shall coincide with the number 

 expressing the order in which it is drawn. 



161. In his first edition Montmort did not give any demon- 

 strations of his results ; but in his second edition he gives two 

 demonstrations which he had received from Nicolas Bernoulli ; 

 see his pages 301, 302. We will take the first of these demon- 

 strations. 



Let a, h, c, d,e, ... denote the cards, n in number. Then the num- 

 ber of possible cases is [n. The number of cases in which a is first 

 is I yi — 1. The number of cases in which h is second, but a not first, 



n — 1 — 1 7i — 2. The number of cases in which c is third, but a 



IS 



not first nor b second, is | w — 1 — | ^^ — 2 — | |?i — 2 — | n — 3 1 



