176 PROFESSOR DE MORGAN, ON THE 



pure time, I should then cite him as an instance of the dogmatism already 

 alluded to : and the more readily, that by the association of the word with 

 his labours, I may claim to have purified it, for the purposes of this paper, 

 from the dyslogistic associations usually connected with it. 



The modern algebraists usually dwell on the second notion, namely that 

 of operation ; and this I shall adopt in the present paper, not only as the 

 most common mode of conception, but also as being equally capable of con- 

 nexion with either of the other two. Imagine the process, whatever it may 

 be, by which we pass from the contemplation of to that of a ; then if a 

 represent a line, we can consider, as a result of our process, either the posi- 

 tion of one extremity with respect to the other, or the quantity of length 

 intercepted between the two. 



I separate the following maxims from the rest as being equally ap- 

 plicable to the symbolical algebra which we have, and to any other 

 which we might have. For it must never be forgotten that, though our 

 present inquiry includes only the possible explanations of one given tech- 

 nical algebra, the subject may and probably must end in the investigation of 

 others, or at least in the extension of the present one. 



1. A simple symbol is the representative of one process, and of one 

 only. 



2. All processes, how many soever, may be looked at in their united 

 effect as one process, and may be represented by one symbol. 



3. Every process by which we can pass from one object of con- 

 templation to another, involves a second by which we can reinstate the 

 first object in its position : or every direct process has another which is 

 its inverse. To complete the separation of these maxims from all others, 

 I propose some considerations connected with the possible extensions of 

 technical algebra. 



The system of explanations which proceeds on the supposition that 

 length affected by direction is the primary object of contemplation in 

 algebra, is well known as to its history by Professor Peacock's Report 

 to the British Association, and as to its present state by the Treatise on 



