256 EULER 



Euler does not demonstrate this result; perhaps he deduced 

 it from the Theory of ProbabiHty itself. For if xr = n, it is 

 obvious that no ticket can be repeated, when all the tickets are 

 drawn in r drawings. Thus the whole number of favourable cases 

 which can occur at the first drawing must be the number of 

 combinations of n things taken r at a time ; the whole number 

 of favourable cases which can occur at the second drawing is the 

 number of combinations of ?2 — r things taken r at a time ; and 

 so on. Then the product of all these numbers gives the whole 

 number of favourable cases. 



This example of the summation of a series indirectly by the aid 

 of the Theory of Probability is very curious ; see also Art. 451. 



461. Euler gives the following paragraph after stating his 

 formulae, 



In his probabilitatibiis aestimandis utique assiimitur omnes litteras 

 ad extrahendum aeque esse proclives, quod autem 111. D^Alemhert negat 

 assumi posse. Arbitratur enim, simul ad omnes tractus jam ante per- 

 actos respici oportere; si enim quaepiam litterae nimis crebro fuerint 

 extractae, turn eas in sequentibus tractibus rarius exituras; contrarium 

 vero eveniie si quaepiam litterae nimis raro exierint. Haec ratio, si 

 valeret, etiam valitura esset si sequentes tractus demum post annum, 

 vel adeo integrum speculum, quin etiam si in alio quocunque loco 

 instituerentur ; atque ob eandem rationem etiam ratio haberi deberet 

 omnium tractuum, qui jam olim in quibuscunque terrae locis fuerint 

 peracti, quo certe vix quicquam absurdius excogitari potest. 



462. In Euler's Opuscula Analytica, Yol. ii., 1785, there is 

 a memoir connected with Life Assurance. The title is Solutio 

 quaestionis ad calculum j)7vhahilitatis pertinentis. Quantum duo 

 conjuges per^solvere deheant, ut suis haeredihus post utriusque 

 mortem certa argenti summa persolvatiir. The memoir occupies 

 pages 315 — 330 of the volume. 



Euler repeats a table which he had inserted in the Berlin 

 Memoirs for 1760 ; see Art. 433. The table shews out of 1000 

 infants, how many will be alive at the end of any given year. 



Euler supposes that in order to ensure a certain sum when 

 both a husband and wife are dead, x is paid down and z paid 



