422 THE COURSE OF MATHEMATICAL STUDIES. 



be evolved from them by innumerable sets of combinations. 

 It does not often occur that even a practised mathematician 

 divines the simplest and the best. His choice among the routes 

 which he may follow is determined by an infinity of circum- 

 stances, and more especially by the way in which the conditions 

 have been expressed. " Words shoot back on the understanding 

 of the wisest," and so do symbols; and if the conditions of the 

 problem, whether geometrical or mechanical, or, if we will, 

 logical, are expressed by means of algebraical symbols, he will, 

 in all probability, not deal with them as he would have done 

 had they been expressed in common language : the reason of 

 which is, that the combinations and inferences which are the 

 most obvious when one mode of expression is employed, cease 

 to be so when it is replaced by another. Hence and from other 

 causes arises a variety of forms of demonstration, often, it is 

 true, perplexing, and yet, if attentively considered, full of in- 

 struction. For as to a mind which has attained to a perfect 

 mastery of the subject, and by which, therefore, the connexion 

 of thejlata of the problem, with its solution is perceived as by 

 intuition, all the demonstrations appear to be in their essence 

 identical, different modes merely of presenting the same con- 

 ceptions, so contrariwise the comparison of the different demon- 

 strations by which a given result has been established, tends to 

 make us recognize the grounds of their essential unity. It is 

 not by merely fixing in the memory the successive steps of a 

 single mode of demonstration, or even by studying several, if 

 we allow them to remain in the mind as distinct and hetero- 

 geneous processes of thought, that we are to acquire a complete 

 insight into the subject in hand, but by a more discursive 

 method, by inquiring perpetually into the grounds and reason 

 of what we are doing, by interpreting our symbols and follow- 

 ing the train of geometrical or physical conceptions to which 

 their interpretation leads, and again by retracing our steps and 

 passing from general considerations or purely geometrical rea- 

 soning to the technical language of symbols. Every change of 

 form should be suggestive of a new aspect of the subject, and 

 it is thus that the simplest way of considering it is to be dis- 

 covered. 



In confirmation of some of the opinions which I have been 

 endeavouring to express, I may refer to Poinsot's admirable 



