CHAPTER II 

 MATHEMATICS 



A STUDY of the first elements and simplest possible 

 principles of mathematics, is then what should first 

 occupy the attention of every would-be student of 

 science. This is absolutely indispensable, since without it 

 no other science is possible ; because all of them, without 

 exception, suppose a greater or less acquaintance with it. 



An objection, however, has sometimes been thus 

 stated. It has been said that mathematics is a most 

 abstract science, and one, therefore, unfitted to occupy 

 the attention of those whose object is to gain a know- 

 ledge of all the concrete, material things about them 

 the things which they can see, feel, and handle. 



Now it is true that the science of mathematics is 

 mainly, and in its simplest branches exclusively, devoted 

 to the study of real or possible numerical relations, 

 apart from the things which bear those relations. 

 Nevertheless common sense shows us at once that 

 numerical relations, or "numbers," can no more exist 

 apart from something which has number, than "weight" 

 can exist apart from something heavy, or "dimension" 

 exist without something or other of a definite size. 

 Numbers, apart from real substantial things, only exist 

 as thoughts, or as the written or spoken signs by which 

 we express numerical relations. But since " numerical 

 relations " have no substantial existence apart from 



