180 DISCOURSE ON THE STUDY 



the case, that in many of the most interesting 

 branches of dynamical enquiry they leave us -com- 

 pletely at a loss. In what relates to the motions of 

 fluids, for instance, this is severely felt. We can 

 include our problems, it is true, in algebraical equa- 

 tions, and we can demonstrate that they contain 

 the solutions ; but the equations themselves are so 

 intractable, and present such insuperable difficulties, 

 that they often leave us quite as much in the dark 

 as before. But even were these difficulties over- 

 come, recourse to experience must still be had, to 

 establish the data on which particular applications 

 are to depend; and although mathematical analy- 

 sis affords very powerful means of representing in 

 general terms the data of any proposed case, and 

 afterwards, by comparison of its results with fact, 

 determining what those data must be to explain 

 the observed phenomena, still, in any mode of 

 considering the matter, an appeal to experience in 

 every particular instance of application is unavoid- 

 able, even when the general principles are regarded 

 as sufficiently established without it. Now, in all 

 such cases of difficulty we must recur to our in- 

 ductive processes, and regard the branches of dy- 

 namical science where this takes place as purely 

 experimental. By this we gain an immense ad- 

 vantage, viz. that in all those points of them where 

 the abstract dynamical principles do afford distinct 

 conclusions, we obtain verifications for our induc- 

 tions of the highest and finest possible kind. When 

 we work our way up inductively to one of these 

 results, we cannot help feeling the strongest assur- 

 ance of the validity of the induction. 



