5 io THE POPULAR SCIENCE MONTHLY. 



chalk-line on a blackboard ; another, as the edge of a knife ; I myself, 

 as the boundary between crystal faces. 



However, in all our minds there is something the same in each. It 

 is the concept or idea. And It is of concepts and ideas that mathemat- 

 ics treats. 



Here Mill seems to make a mistake. He says, " The points, lines, 

 circles, and squares which any one has in his mind are simply copies 

 of the points, lines, circles, and squares which he has known in his 

 experience." To his mind, then, the function of thought, when we 

 think of circles, is to reproduce some original sensation more or less 

 vividly. This, however, is what I call imagination ; and we have tried 

 to prove that imaginations were not the objects of mathematical treat- 

 ment. Helmholtz acknowledges this when he says that the axioms of 

 geometry, taken by themselves out of all connection with mechanical 

 propositions, represent no relations of real things. 



We will notice certain other facts about concepts and words, in 

 connection with their mathematical relations. The first is the per- 

 sistence of concepts. By this I mean that an idea once formed, by 

 whatever means, experimental or otherwise, does not depend upon the 

 continual recurrence of the same experience for its continued existence. 



That is, having once formed an idea of a baby hippopotamus, by 

 having seen one in Barnum's Great Show, I have that idea, which is 

 called into use on various other occasions such as hearing of it in the 

 newspapers. It is not at all necessary that I should renew the experi- 

 ence every time Barnum comes around. It is, of course, true that a 

 concept may be disused, but its use may be made common as well by 

 unlike as by like experiences. 



However, on closer inspection of the hippopotamus, my conception 

 may be new. This leads us to our other all-important distinction and 

 division. Every name has a denotation and a connotation. Its de- 

 notation is usually of things, its connotation is conceptual. Some 

 words, proper names especially, correspond to things, the ideas attached 

 to which vary according to the varying aspect of the thing. Other 

 words, however, correspond to ideas ; these words are applied or not 

 to things according as there are experiences coming under the concept 

 to which they are attached. 



This distinction between words with fixed denotation and varying 

 connotation and words of fixed connotation and varying denotation is 

 quite important, as we shall see. Let us first, however, return to our 

 hippopotamus. This is a word for me of at least partially fixed deno- 

 tation ; it must include the animal that I saw ; it must not include 

 an ordinary pig. The connotation would be almost indefinite. This 

 word has, then, a fixed denotation varying connotation, approximately. 

 On the other hand, take the name, rigid body. This is a name with 

 a denotation varying down to zero, perhaps, but its connotation is 

 changeless. 



