30 ARITHMETICAL RECBEATIONS [CH. II 



Sixth Fallacy. The following demonstration depends on 

 the fact that an algebraical identity is true whatever be the 

 symbols used in it, and it will appeal only to those who are 

 familiar with this fact. We have, as an identity, 



\x — y — i \/y — x (i), 



where i stands either for + V— 1 or for — V— 1. Now an 

 identity in x and y is necessarily true whatever numbers x and 

 y may represent. First put x = a and y = b, 



.'. \fa — b = i^b — a (n). 



Next put x = b and y = a, 



.". V& — a = i<Ja — b (iii). 



Also since (i) is an identity, it follows that in (ii) and (iii) the 

 symbol i must be the same, that is, it represents + V— 1 or 

 — V— 1 in both cases. Hence, from (ii) and (iii), we have 



Va-6 Vb — a = i 1 V& - a >Ja — b, 

 .'. 1 = i\ 

 that is, 1 = — 1. 



Seventh Fallacy. The following fallacy is due to D'Alem- 

 bert*. We know that if the product of two numbers is equal 

 to the product of two other numbers, the numbers will be in 

 proportion, and from the definition of a proportion it follows 

 that if the first term is greater than the second, then the third 

 term will be greater than the fourth: thus, if ad = bc, then 

 a:b = c:d, and if in this proportion a > b, then c> d. Now if 

 we put a = d = 1 and b = c = — 1 we have four numbers which 

 satisfy the relation ad = be and such that a > b ; hence, by the 

 proposition, c> d, that is, — 1 > 1, which is absurd. 



Eighth Fallacy. The mathematical theory of probability 

 leads to various paradoxes : of these one specimen - } - will suffice. 

 Suppose three coins to be thrown up and the fact whether each 

 comes down head or tail to be noticed. The probability that 

 all three coins come down head is clearly (1/2) 8 , that is, is 1/8 j 



• Opuscules Mathimatiques, Paris, 1761, vol. i, p. 201. 



+ See Nature, Feb. 16, March 1, 1894, vol. xlix, pp. 365— 3G6, 413. 



