MODERN SCIENCE. 197 



ways, different in length, but identical in their final results, 

 of getting from an angular point of a triangle to another " 

 (Jevons). He also pointed out that "the three symbols of 

 ordinary algebra form but three of an interminable series of 

 conceivable systems." 



The foregoing sketch of the work done by mathematicians, 

 the present writer candidly owns, is here both meagre and 

 unsatisfactory. It gives a list of new instruments of calculation 

 it is true ; but it neither defines the object for which they 

 were created, nor states what discoveries were made by the 

 use of them ; so that the general reader who may not be 

 conversant with mathematical science can form no estimate 

 of the value of the facts just enumerated. But as the two 

 next sections will describe the power of mathematics in 

 solving difficulties in astronomy and physics which would be 

 insuperable without it, the reader will have occasion then to 

 appreciate its effects in a multitude of facts. He will see the 

 imperial sway of mathematics over all other branches, and 

 will plainly understand why it is often called the Queen of 

 Sciences. The names of the mathematicians, which may be 

 names and nothing more to him, will then appear illumined 

 by a halo suddenly revealing all the splendours of genius, for, 

 indeed, in the fame-roll of all times they hold the first rank. 



In order to give an idea of the vastness of modern science, 

 a remark may parenthetically be made here, based on a fact 

 of importance. 



Mathematics, if considered in detail, embrace pure mathe- 

 matics and applied mathematics : 



/ I. ARITHMETIC, subdivided into 



A. Pure Arithmetic : 



(a) Theory of Numbers, (b} Ratios, 



(c] Proportions, (d) Scales of Notation. 



B. Applied Arithmetic : 



(a) Weights, (ff) Measures. 



II. ALGEBRA, into 



A. Elementary : 



(a) Analysis of Determinate Quantities, 

 (3) Analysis of Indeterminate Quantities. 



B. Higher Infinitesimal Calculus : 



(a) Differential, () Integral. 



