21 



appertains in a greater or less degree to all other 

 Sciences, and perhaps also to all pursuits in life. In 

 its highest flight each soars to a distance from its 

 felloAVS. Each is pursued alone for its own sake, and 

 without reference to its connection with, or its applica- 

 tion to, any other subject. The pioneer and the advanced 

 guard are of necessity separated from the main body ; 

 and in this respect Mathematics does not materially 

 differ from its neighbours. And, therefore, as the 

 solitariness of Mathematics has been a frequent theme 

 of discourse, it may be not altogether unprofitable to 

 dwell for a short time upon the other side of the 

 question, and to inquire whether there be not points 

 of contact in method or in subject-matter between 

 Mathematics and the outer world which have been 

 frequently overlooked ; whether its lines do not in some 

 cases run parallel to those of other occupations and 

 purposes of life ; and lastly, whether we ma;f not hope 

 for some change in the attitude too often assumed 

 towards it by the representatives of other branches of 

 knowledge and of mental activity. 



In his Preface to the Principia, Newton gives expres- 

 sion to some general ideas which may well serve as tho 

 key note for all future utterances on the relation of 

 Mathematics to Natural, including also therein what are 

 commonly called Artificial^ Phenomena. 



"The ancients divided Mechanics into two parts, Newton'^ 

 " Rational and Practical; and since artizans often work P^®^^®- 

 *' inaccurately, it came to pass that Mechanics and 

 " Geometry were distinguished in this way, that every- 

 " thing accurate was referred to Geometry, and every- 

 " thing inaccurate to Mechanics. But the inaccuracies 

 " appertain to the artizan and not to the art, and 

 " Geometry itself has its foundation in mechanical 



