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CHAPTER II. 

 ARITHMETICAL RECREATIONS CONTINUED. 



I devote this chapter to the description of some arithmetical 

 fallacies, a few additional problems, and notes on one or two 

 problems in higher arithmetic. 



Arithmetical fallacies. I begin by mentioning some 

 instances of demonstrations* leading to arithmetical results 

 which are obviously impossible. I include algebraical proofs as 

 well as arithmetical ones. Some of the fallacies are so patent 

 that in. preparing the first and second editions I did not think 

 such questions worth printing, but, as some correspondents 

 expressed a contrary opinion, I give them for what they are 

 worth. 



First Fallacy. One of the oldest of these — and not a very 

 interesting specimen — is as follows. Suppose that a = b, then 



ab = a\ .-. ab-b* = a t -b>. .". b(a-b) = (a + b)(a-b). 

 .-. b = a + b. .-. 6 = 26. .-. 1 = 2. 



• Of the fallacies given in the text, the first and second are well known ; 

 the third is not new, but the earliest work in which I recollect seeing it is 

 my Algebra, Cambridge, 1890, p. 430 ; the fourth is given in G. Chrystal's 

 Algebra, Edinburgh, 1889, vol. n, p. 159 ; the sixth is due to G. T. Walker, 

 and, I believe, has not appeared elsewhere than in this book; the seventh 

 is due to D'Alembert; and the eighth to F. Galton. It may be worth re- 

 cording (i) that a mechanical demonstration that 1=2 was given by E. Chartres 

 in Knowledge, July, 1891; and (ii) that J. L. F. Bertrand pointed out that 

 a demonstration that 1=-1 can be obtained from the proposition in the 

 Integral Calculus that, if the limits are constant, the order of integration is N 

 indifferent ; hence the integral to x (from x=0 to a;=l) of the integral to y (from 

 j/ = to i/ = l) of a function should be equal to the integral to y (from y = to 

 y = l) of the integral to x (from i=0 to x=l) of <j>, but if <p={x 3 -y i )l(x''+y'') 2 , 

 this gives £t= - Jir. 



