THE COURSE OF MATHEMATICAL STUDIES. 419 



in order to this kind of mental development, it is necessary not 

 only that the Student should master the successive steps of the 

 demonstrations set before him and retain them in his memory, 

 but that his mind should become imbued with their spirit and 

 essence. His real progress therefore is not to be measured 

 simply by the extent of ground over which he has passed : it 

 varies also according to the degree in which he has approached 

 towards a complete intuition into the results which he is able to 

 prove. 



I believe that this principle ought to be our guide in ex- 

 amining the merits and defects of a course of mathematical study 

 intended to form part of a liberal education. But the connexion 

 of natural philosophy with mathematics must, to a greater or 

 less extent, modify the conclusions to which it would lead us. 



II. It would be impossible to trace in detail the conse- 

 quences which appear to follow from this way of considering 

 the subject. They may be classed under two heads, the choice 

 of subjects, and the choice of methods. With respect to the 

 former, my impression has long been that a good deal might be 

 omitted which now enters into our course of reading, not only 

 without impairing its utility, but with positive advantage. A 

 more rigorous subordination of details to fundamental principles 

 would not only save the Student's time, but would make the 

 principles themselves be more clearly apprehended. Everything 

 received into our course ought to justify its admission there, 

 either by its own importance or by its connexion with something 

 more valuable than itself. Mere exercises of industry and in- 

 genuity, long numerical calculations, complicated processes of 

 algebraical reduction, tricks of transformation for the evaluation 

 of integrals and the solution of differential equations, and the 

 like, may all be accounted comparatively useless. These things 

 may be impressed on the memory but will hardly long remain 

 there, and meanwhile are felt to be rather a burden than an 

 acquisition. So in mixed mathematics, many of the approxi- 

 mate formulae in optics, the less important astronomical cor- 

 rections, detailed descriptions of philosophical instruments, &c., 

 might all be advantageously laid aside. Not that these things 

 are not worth knowing, but that they do not properly belong 

 to such a course of mathematics as we are considering in which 

 the chief end proposed is a clear insight into fundamental priri- 



272 



