MATHEMATICS 39 



quantities themselves. With such matters the highest 

 branches of mathematics are concerned, but they are, of 

 course, quite beyond the range of an introduction to the 

 elements of science. 



For information concerning all but the rudiments of 

 mathematics, the reader is referred to the various works 

 specially devoted to the teaching of that science. 



But before concluding this chapter we desire again to 

 insist on the correspondence which exists between the 

 truths of mathematics, of whatever order, and the proper- 

 ties of real substantial things. The truths of geometry are 

 made evident to the eye by diagrams and to the mind by 

 reasoning. The truths of algebra may, as before said,* 

 be tested by taking certain numbers as exponents of the 

 algebraic signs, and so reducing algebra to arithmetic, 

 while the truths of arithmetic may be demonstrated by 

 the seeing and handling of corresponding numbers of 

 real material bodies. We may now pass on to the study 

 of elementary truths, second only to mathematics in the 

 universality of their application. 



* See ante, p. 33. 



