420 THE COURSE OF MATHEMATICAL STUDIES. 



ciples. In general it may be said that formulae of approxima- 

 tion are unsuited to the end we have in view ; they give little 

 or nothing on which the mind can rest : their value resides in 

 the practical application which is to be made of them, but which 

 the Student never makes. It is the predominance of approxi- 

 mate results which renders the lunar and planetary theories 

 unsatisfactory portions of the Student's course. They are, how- 

 ever, by no means to be omitted or curtailed, and with respect 

 to the latter, the evil might be lessened, though not without 

 some inconvenience, by giving more prominence to the general 

 theory of the variation of parameters as we find it in the Me- 

 canique Analytique and in some of Poisson's memoirs. Sir W. 

 Hamilton's essays in the Philosophical Transactions, and those 

 of Jacobi in Crelle's Journal, might, perhaps, give some ad- 

 ditional materials for the formation of a course of study on this 

 part of natural philosophy. 



III. Secondly, as to the choice of methods, and especially 

 as to the preference to be given to geometry or to analysis. 

 Ever since the introduction of the modern analysis into Univer- 

 sity reading, there have been complaints of its having super- 

 seded the older methods and traditions of the Cambridge system. 

 Those who favoured its progress affirmed, and most truly, that 

 by its aid the Student advances faster, and goes farther, than 

 he could do without it ; he gains in fact more knowledge of the 

 subjects set before him. But this argument had little weight 

 with those who held that not the knowledge but the process of 

 acquiring it the training and discipline of the mind was the 

 thing chiefly to be thought of. 



It has been said that if information merely is the end in 

 view, mathematics have less claim on our attention than many 

 other things, and that most of the arguments in their favour 

 cease to be applicable if geometry is discarded or disparaged. 

 Of late these views seem to have gained ground in the Univer- 

 sity : their influence may be traced in the recent legislation on 

 the subject of mathematical honours. The principle on which 

 this re-action against the newer methods is chiefly based, namely, 

 that the mind of the Student ought to be as much as possible 

 conversant with fundamental conceptions is, I think, perfectly 

 correct. But it does not follow that analytical methods ought 

 to be discouraged. Demonstrations may be geometrical, arid 



