346 Fallacies. [CHAP, xiv. 



To suggest to any individual or company that they 

 should consent to go- on playing upon such terms as these 

 would be too barefaced a proposal. And yet the case in 

 question is identical in principle, and almost identical in 

 form, with this. To offer to give a man any sum he likes to 

 name provided he gives you back again that same sum plus 

 one, and to offer him any number of terms he pleases of the 

 series 1, 2, 4, 8, 16, &c., provided you have the next term of 

 the set, are equivalent. The only difference is that in the 

 latter case the result is attained with somewhat more of 

 arithmetical parade. Similarly equivalent are the processes 

 in case we prefer to leave it to chance, instead of to choice, 

 to decide what sum or what number of terms shall be fixed 

 upon. This latter is what is really done in the case in 

 question. A man who consents to go on doubling his stake 

 every time he wins, is leaving nothing else to chance than 

 the determination of the particular number of terms of such 

 a geometrical series which shall be allowed to pass before he 

 stops. 



15. It may be added that there is no special virtue in 

 the particular series in question, viz. that in accordance with 

 which the stake is doubled each time. All that is needed is 

 that the last term of the series should more than balance all 

 the preceding ones. Any other series which increased faster 

 than this geometrical one, would answer the purpose as well 

 or better. Nor is it necessary, again, that the game should 

 be an even or fair one. Chance, be it remembered, affects 

 nothing here but the number of terms to which the series 

 attains on each occasion, its final result being always arith 

 metically fixed. When a penny is tossed up it is only on 

 one of every two occasions that the series runs to more than 

 two terms, and so his fixed gains come in pretty regularly. 

 But unless he was playing for a limited time only, it would 



