DEVELOPMENT OF MATHEMATICAL THOUGHT. 687 



as instruments or devices for the solution of definite 

 problems in arithmetic, geometry, and mechanics. The 

 solution of the equation i.e., the expression of the un- 

 known quantity in terms of the known quantities 

 served a practical end. Gradually as such solutions be- 

 came more and more difficult, owing to the complexity of 

 the formulae, the doctrine divided itself into two distinct 

 branches, serving two distinct interests. The first, and 

 practically the more important one, was to devise methods 

 by which in every single case the equations which 

 presented themselves could be solved with sufficient 

 accuracy or approximation ; this is the doctrine of the 

 numerical solution of equations. The other more scien- 

 tific branch looked upon equations as algebraical ar- 

 rangements of quantities and operations which possessed 

 definite properties, and proposed to investigate these 

 properties for their own sake. The question arose, 

 How many solutions or roots an equation would admit of, 

 and whether the expression of the unknown quantity in 

 terms of the known quantities was or was not possible 

 by using merely such operations as were indicated by 

 the equation itself i.e., the common operations and the 

 ordinary numbers of arithmetic ? This doctrine of the 

 general properties of equations received increasing atten- 

 tion as it became empirically known that equations 43. 



General 



beyond the fourth degree could not be solved in the solution of 



equations. 



most general form. 1 Why could they not be solved, 



1 Since the researches regard- 

 ing the solubility of Equations 

 have led on, through Galois and 

 the French analysts, to the same 

 line of reasoning as other re- 

 searches mentioned before viz., 



toward the development of the 

 theory of groups the history of 

 the whole subject has aroused 

 special interest. The earlier be- 

 ginnings and the labours of for- 

 gotten analysts have been un- 



