644 



SCIENTIFIC THOUGHT. 



12. 



Complex 

 quantities. 



13. 

 The con- 

 tinuous. 



14. 



The infinite. 



the last hundred years has grown in proportion to the 

 methodical study and stricter definition of the notions of 

 the complex quantity, of the continuous, and of the infinite. 

 And these conceptions indicate three important logical 

 developments which characterise modern mathematical 

 reasoning. The conception of the complex quantity or 

 the complex unit introduces us to the possible extension 

 of our system of counting and measuring, retaining or 

 modifying, the fundamental rules on which it is based. 

 The conception of the continuous and its opposite, the 

 discontinuous, introduces us to the difference of numbers 

 and quantity, numbers forming a discontinuous series, 

 whilst we conceive all natural changes to be made up of 

 gradual — i.e., of imperceptibly small — changes, called by 

 Newton fluxions. The discussion, therefore, of the con- 

 tinuous leads us ultimately to the question how our 

 system of counting can be made useful for dealing with 

 continuously variable quantities — the processes of nature. 

 The conception of the infinite underlies not only the 

 infinitesimal methods properly so called, but also all the 

 methods of approximation by which — in the absence of 

 rigorous methods — mathematical, notably astronomical, 

 calculations are carried out. 



Problems involving one or more of these concep- 

 tions presented themselves in large number to the 

 analysts of the eighteenth century : there were notably 

 two great doctrines in which they continually occur — 

 the general solution of equations,^ and the theory of 



^ A.S it may not be immediately 

 evident how the ideas of continuity 

 have to do with the general solution 

 of equations, I refer to the first 



publication by Gauss, in 1799, con- 

 taining a proof of the fundamental 

 theorem of algebra, and its republi- 

 cation fifty years later (see Gauss, 



