X. On the Foundation of Algebra. By Augustus De Morgan, 

 F.R.A.S. F.C.P.S.; of Trinity College; Professor of Mathematics 

 in University College, London. 



[Read Dec. 9, 1839.] 



The extent to which explanation of the meaning of the symbolical 

 results of Algebra has been carried within the last half century ; the com- 

 plete interpretation of all which formerly appeared incongruous; the sepa- 

 ration, as it was called, of the symbols of operation and quantity, which 

 amounts to the use of an algebra in which the symbols represent something 

 more than simple magnitude ; — will for some time to come suggest inquiry 

 into the logic of this many-handled instrument of reasoning, which seems 

 to be capable of presenting, under fixed laws of operation, all the results 

 which arise from very distinct primary conceptions as to the things 

 operated upon. 



When several different hypotheses lead to results which admit of a com- 

 mon mode of expression, we are naturally led to look for something which 

 the hypotheses have in common, and upon which the sameness of the method 

 of expression depends. A comparison of the properties of the ellipse and 

 hyperbola would bewilder the imagination, under any of the distinct defi- 

 nitions which might be given of the two curves ; nor would the mind 

 rest satisfied until it had discovered the reason of the similarity which exists 

 between these properties. 



Algebra now consists of two parts, the technical, and the logical. Tech- 

 nical algebra is the art of using symbols under regulations which, when this 

 part of the subject is considered independently of the other, are prescribed 

 as the definitions of the symbols. Logical algebra is the science which 

 investigates the method of giving meaning to the primary symbols, and of 

 interpreting all subsequent symbolic results. It is desirable that the word de- 

 finition should not enter in two distinct senses, and I should propose to retain 



