MONTMORT. 115 



comprendre comment on pent determiner la quantite de vieille ma- 

 tiere qui reste dans nos corps melee avec de la nouvelle qui nous 

 vient tous les jours par la nourriture, pour reparer la perte que nos 

 corps font insensiblement par la transpiration continuelle. 



The dissertation De Nutritione will be found in the collected 

 edition of John Bernoulli's works ; see Vol. I. page 275. 



198. John Bernoulli passes on to a remark on Montmort's 

 discussion of the game of Treize. The remark enunciates the 

 following theorem. 



Let <^(.) = l--^+^-^+...+-^ 



and let 



n 



t(«)=^(«) + J^("-l) + U(n-2)+...+ ^^<^(1); 



111 1 



then shall i/r {n) = --[-+_++. ..+_. 



^ [^ [I [± \JL 



We may prove this by induction. For we may write yjr (n) in 

 the following form, 



T fl 1 1 1 1 1 



^11 



2 ] ^ ' 1 ' 12 ' [3_ 



1 r, 1 1 1 1 1 



-o^ l + T + n5 + rT. + + ^^3^ J 



n 





Hence we can shew that 



y}r (n + 1) = i/r {n) + 



;i + 1 * 



199. John Bernoulli next adverts to the solutions which 

 Montmort had given of the five problems proposed by Huygens ; 

 see Art. 35. 



According to John Bernoulli's opinion, Montmort had not 

 understood the second and third problems in the sense which 

 Huygens had intended ; in the fifth problem Montmort had 



8-^2 



