120 THE POPULAR SCIENCE MONTHLY 



may be set for him by the physicist or the engineer. If he had done 

 this in the past he would not have created the instruments necessary to 

 solve such problems, and hence it is unreasonable to make such re- 

 strictions as to the future. 



If the physicists of the eighteenth century had abandoned the study 

 of electricity because it seemed to serve no useful end, we should not 

 have had the many useful applications of electricity during the nine- 

 teenth century. Similarly, if the mathematician had abandoned the 

 study of negative and imaginary numbers because they seemed to 

 point only to impossibilities, we should not have had the many power- 

 ful instruments of thought which enable us to cope more successfully 

 with many problems of nature. Just as the physicist is largely guided 

 in his work by those facts which seem to point to general laws, so 

 the mathematician is guided in his work by the desire to discover ex- 

 tensive relations and laws having a wide range of application. Millions 

 of isolated facts present themselves to the investigator, some of which 

 are of striking interest to the initiated, but they are of practically no 

 value in the development of mathematics except that they may some- 

 times serve as an exercise in secondary instruction. 



At a first thought the statement that " Mathematics is the art of 

 giving the same name to different things " may appear to be entirely 

 contrary to fact, but from a certain standpoint this statement conveys 

 a very fundamental truth. It should be borne in mind that these dif- 

 ferent things must have in common the property to which this com- 

 mon name refers, and that it is the duty of the mathematician to 

 discover and exhibit this common property. By way of illustration 

 we may recall the use of x for various unknowns in algebra and the 

 (1,1) correspondence between the two series of operators. When the 

 language has been properly chosen it is often surprising to find that 

 the demonstrations, as regards a known object, apply immediately to 

 a large number of new objects without even a change of name. 



Just as the boundaries between the elementary subjects of mathe- 

 matics — arithmetic, algebra and geometry — vanish when the knowledge 

 of these subjects is sufficiently extended, so the boundaries between 

 subjects in pure and applied mathematics are disappearing, and it is 

 exactly in these bordered lands, or in this common territory of two or 

 more subjects, where the greatest recent progress has been made and 

 where the greatest future activity may be expected. The work in this 

 common territory is made possible by observing similarity of form where 

 there is dissimilarity of matter, or by observing some other common 

 properties which admit mathematical treatment. 



In Poincare's address some of these general observations were illus- 

 trated by numerous examples chosen from various fields of higher 

 mathematics. On the contrary, we shall confine our illustrative ex- 



