CH. II] ARITHMETICAL RECREATIONS 29 



Second Fallacy. Another example, the idea of which is due 

 to John Bernoulli, may be stated as follows. We have (- 1) 2 = 1. 

 Take logarithms, . ■ . 2 log (- 1) = log 1 = 0. . • . -log (- 1) = 0. 

 .-. -l = e°. .-. -1 = 1. 



The same argument may be expressed thus. Let # be a 

 quantity which satisfies the equation e*= — 1. Square both 

 sides, 



.-. e w =l. .-. 2a = 0. .-. x = 0. .-. e*=e\ 



But e*=-l and e°=l, .-. -1 = 1. 



The error in each of the foregoing examples is obvious, but 

 the fallacies in the next examples are concealed somewhat 

 better. 



Third Fallacy. As yet another instance, we know that 



log (1 +#) = #- \ot? + %a?—.... 

 If x=l, the resulting series is convergent; hence we have 

 lo g 2 = l-i + £-i + £-£ + *-£+i-.... 

 .-. 21og2 = 2-l + S-£ + f-$ + ?-i + $-.... 

 Taking those terms together which have a common denominator, 

 we obtain 



21og2 = l+£-£ + £ + f-i + £-. 



= l-i + i-i + £- 



= lo g 2. 



Hence 2 = 1. 



Fourth Fallacy. This fallacy is very similar to that last 

 given. We have 



log2 = l-4 + £-i + W + - 



= 0+i+£+-)-(i + i + i + -) 



= {(i+i + H-)+(i+i+i + -)}-2(£ + m+...) 

 = {i+i+i + ...}-(i + i+i + ...) 



= 0. 

 Fifth Fallacy. We have 



•J a x 4h = Va&. 

 Hence V^I x V^T = V(-l)(-l), 



therefore, (V^Tj^Vl, that is, -1 = 1. 



