290 PROFESSOR DE MORGAN, ON THE 



the countries painted on them into one of a more particular kind by 

 help of the forms of the pieces, more resembles the investigator and the 

 mathematician. 



Mr. Warren, in his explanation* and Dr. Peacock, in his interpre- 

 tation*, of the algebraical symbols, have both obtained the symbols 

 a + b and ab independently of each other as to their meaning : while 

 both, to obtain the meaning of the symbol a b , have had recourse to the 

 fundamental derivation from a, aa, aaa, &c. The consequence is, that 

 while both establish most completely the ordinary forms of algebra, 

 neither is prepared to consider a h where b is other than what answers 

 to the positive or negative number or fraction of the semi-logical algebra. 

 Mr. Warren, who carefully avoids all interpreted results, and whose work 

 is as complete a succession of consequences from explanations adequate 

 to the results as that of any professedly arithmetical algebraist, has there- 

 fore totally avoided the use of such a symbol as e e V _1 , using instead 



/ \ — 



( 1 ) 2,r > a new convention, derived from the roots of unity. Dr. Pea- 

 cock, making use virtually of the equation 



cos + J- 1 sin 9 = (cos 1 + J- J sin l) fl , 



and denoting cos 1 + J — l sin 1 by e, is able fully to interpret all results 

 arising from e e = cos 9 + J — 1 sin 9, and to prove this equation as a 

 consequence of the laws of operation. Both writers would consider 

 e^V-i = cos 6 + J — I sin 9, as an equation resembling — (— A) = A in 

 ordinary algebra, of which the first side, known to be the same symbol 

 as the second, can only receive its explanation from the second. And 

 we see that the complete independence of the explanations or interpre- 

 tations of a + b and ab leads (in the works alluded to) to a full and 

 satisfactory account of their properties, while the derivation of a b from 

 ab ends in a partial and insufficient notion of the meaning of the symbol 

 itself. There is something disappointing in the first-mentioned circum- 

 stance, since the mind naturally looks, in the most extensive view of the 

 subject, for the prototypes of those analogies and modes of derivation 

 which were of such essential use in the more bounded science : but at the 



* I use these words in the same sense as in my last Paper. 



