326 JOHN BERNOULLI. 



abandonner mon dessein, et je me reservai seulement de voir par le 

 Memoire de cet illustre Geometre si j'avois raisonue juste; il a eu la 

 bonte de me le commuDiquer et j'ai vu que le peu que j'avois fait, etoit 

 fonde sur des raisonnemens qui, s'ils n'etoient pas sublimes, n'etoient du 

 moins pas faux. 



600. Jobn Bernoulli does not give an Algebraical investiga- 

 tion ; lie confines himself to the arithmetical calculation of the 

 chances of the various kinds of sequences that can occur when 

 there are 90 tickets and 2 or 3 or 4 or 5 are drawn. His method 

 does not seem to possess the advantage of facility, as compared 

 with those of Euler and Beguelin, which he himself ascribes to it. 



601. There is one point of difference between John Bernoulli 

 and Euler. John Bernoulli supposes the numbers from 1 to 90 

 ranged as it were in a circle ; and thus he counts 90, 1 as a 

 binary sequence ; Euler does not count it as a sequence. So also 

 John Bernoulli counts 89, 90, 1 as a ternary sequence ; with Euler 

 this would count as a binary sequence. And so on. 



It might perhaps have been anticipated that from the greater 

 symmetry of John Bernoulli's conception of a sequence, the in- 

 vestigations respecting sequences would be more simple than on 

 Euler's conception ; but the reverse seems to be the case on ex- 

 amination. 



In the example of Art. 440 corresponding to Euler's results 



o / ON/ ON (n-2) (n-S) {n-4<) 

 n-2, {n - 2) (n - 3), -^^ ^ ^ ^ g , 



we shall find on John Bernoulli's conception the results 



602. There is one Algebraical result given which we may 

 notice. Euler had obtained the following as the chances that there 

 would be no sequences at all in the case of n tickets ; if two 



tickets be drawn the chance is , if three -^^ ^'-W^ — , if 



n n{n — l) 



four ( ^-4)(^-5)(n-6) (^,5)(.^ -6)(^-7) (/^-8) . 



n{n-\){n-2) ' " "^^ n{n-l) {n-2) {n-^) ' 



and so the law can be easily seen. Now John Bernoulli states 



