EULER. 253 



454. With the notation of Finite Differences we may denote 

 the number of cases favourable to the drawing of m specified 

 tickets by A'" {(/> (n — 7?^, r)}^; and the number of cases favourable 

 to the drawing of all the tickets by A" {(/> (0, r)Y. 



455. In the Histoire de VAcad. ... Paris, 1783, Laplace gives 

 an approximate numerical calculation, which also occurs in 

 page 195 of the Theorie ... des Proh. He finds that in a lottery 

 of 10000 tickets, in which a single ticket is drawn each time, it 

 is an even chance that all will have been drawn in about 957()7 

 drawings. 



456. After this notice of what had been published by De 

 Moivre and Laplace, we proceed to examine Euler's solution. 



The problem appears in Euler's Opuscida Analytica, Vol. Ii., 

 1785. In this volume pages 331 — 346 are occupied with a memoir 

 entitled Solutio quarundam quaestionum dijjiciliorum in calculo 

 prohabilium. Euler begins thus : 



His quaestionibus occasionem dedit ludus passim publice institutus, 

 quo ex nonaginta scliedulis, numeris 1, 2, 3, 4,... 90 signatis, statis tem- 

 poribus quinae schedulae sorte extrahi sclent. Hinc ergo hujusmodi 

 quaestiones oriuntur: quanta scilicet sit probabilitas ut, postquam datus 

 extractionum numerus fuerit peractus, vel omnes nonaginta numeri 

 exierint, vel saltern 89, vel 88, vel pauciores. Has igitur quaestiones, 

 utpote difficillimas, hie ex principiis calculi Probabilium jam pridem usu 

 receptis, resolvere constitui. Neque me deterrent objectiones Illustris 

 lyAlembert, qui huuc calculum suspectum reddere est conatiis. Post- 

 quam enim summus Geometra studiis mathematicis valedixit, lis etiam 

 helium indixisse videtur, dum pleraque fundamenta solidissinie stabilita 

 evertere est aggressus. Quamvis enim hae objectiones apud ignaros 

 maximi ponderis esse debeant, hand tamen metuendum est, inde ipsi 

 scientiae ullum detrimentum allatum iri. 



457. Euler says that he finds a certain symbol very useful in 

 these calculations ; namely, he uses 



_q] 1.2 q 



458. Euler makes no reference to his predecessors De Moivre 

 and Laplace. He gives the formula for the chance that all the 



