THE COURSE OF MATHEMATICAL STUDIES. 421 



yet in a high degree artificial ; and first principles may be lost 

 sight of in a maze of triangles, no less than in a maze of equa- 

 tions. Though in mathematical investigations there is no royal 

 road, yet there is a natural one, that, namely, which enables the 

 Student, as far as possible, to grasp the natural relations which 

 exist among the objects of his contemplation. If this route be 

 followed, it matters but little whether the reasoning be expressed 

 by one set or kind of symbols or by another in plain words 

 in short hand or algebraically. To -change the notation is 

 merely to translate from one language into another. 



It is common to find persons in Cambridge and elsewhere 

 who insist upon it that geometry is geometry, and analysis 

 analysis ; but it may be doubted whether this notion of an 

 absolute separation between the two things is not the result of 

 a want of familiarity with either. It seems to be supposed that 

 if a mathematician treats a problem geometrically, he has to 

 think about it for himself, whereas if he treats it symbolically, 

 the symbols think for him. Perhaps it may be said that the 

 fact of there being any tendency towards so childish a notion is 

 in itself evidence of the mischief produced by the use of sym- 

 bols ; and certainly if symbols were never used, the notion could 

 not exist. But neither could it exist if they were rightly used 

 and rightly understood. The phrases which I believe may now 

 and then be heard from some of our younger analysts, such as 

 " the irrefragable a?," and " putting it into the mill," for ex- 

 pressing the conditions of a problem symbolically, show perhaps 

 that those who use them have but a half understanding of what 

 they are doing. But this evil is not to be remedied by discou- 

 raging the use of symbols. That our methods should be geo- 

 metrical is not by any means essential ; they ought to be natural, 

 and it has been too hastily supposed that they will necessarily 

 be so if symbols are excluded : whereas it is not by precise 

 adherence to any particular mode of expression that we are to 

 bring the Student to a familiar apprehension of the principles 

 of what he is engaged on. This is to be accomplished rather 

 by a " melange heureux de synthese et d'analyse," to use the 

 words of a great master in the art of which he speaks, than by 

 imposing either on teacher or students any unnecessary re- 

 straints. Let us consider the question more generally. When 

 the conditions of a problem have been stated, the solution may 



