174 PROFESSOR DE MORGAN, ON THE 



it as used in the art of algebra, applying the terms explanation and interpreta- 

 tion to denote the preparatory and terminal processes of the science. Thus a 

 symbol is defined when such rules are laid down for its use as will enable us 

 to accept or reject any proposed transformation of it, or by means of it. A 

 simple symbol is explained when such a meaning is given to it as will enable 

 us to accept or reject the application of its definition, as a consequence of 

 that meaning: and a compound symbol is interpreted, when, having 

 occurred as a result of explained elements, used under prescribed defini- 

 tions, a necessary meaning can be given to it ; the necessity arising from the 

 tacit supposition that the compound symbol, considered as a new simple 

 one, must still be subject to the prescribed definitions, when it subsequently 

 comes in contact with other symbols. The last words may need the remark, 

 that though we sometimes appear to interpret a symbol merely for the 

 purpose of explaining a result, ye we know that such interpretation would 

 be subsequently rejected, if the use of the symbol, under the prescribed 

 definitions, were not found to be logically admissible. 



A symbol is not the representation of an external object absolutely, but 

 of a state of the mind in regard to that object ; of a conception formed, for 

 the formation of which the mind knows that it is or was indebted to the 

 presence, bodily or ideal, of the object. Those who do not remember this, 

 the real use of a symbol, are apt to dogmatize* declaring one or another 

 explanation of a symbol, that is, the signification by it of one or another 

 impression produced on their own minds, to be real, true, natural, or neces- 

 sary : it being neither one nor the other, except with reference to the par- 

 ticular mind in question. To take a very simple case, and one which bears 

 upon our subject, let us imagine that we form successively a conception of 

 the absence of all definite magnitude, followed by one of the existence of a 

 certain magnitude, say a line of given length. The mind of one person may 

 pass from the one to the other by imagining the given length to be instanta- 

 neously generated, no one portion of it coming into the thoughts before 

 or after another ; that of a second may make the transition by imagining a 

 point to move from one extremity to the other : while that of a third may 

 dwell rather on the relative position of the two extremities, and may think 



* Of course, I use this word in its primitive sense, without any censure implied : the very- 

 sentence in which the word occurs is, and is meant to be, dogmatical. 



