THE COURSE OF MATHEMATICAL STUDIES. 423 



tract on the motion of a rigid body. He lias there shown, with 

 great felicity both of thought and of expression, that the art of 

 combining symbols is by no means the whole of mathematical 

 analysis ; that we must join to it the art of interpretation, and 

 that in many cases the essence and meaning of a result are 

 scarcely more obvious in the equations which express it than in 

 the original "mise en equation." 



It did not belong to his purpose to point out that on the 

 other hand geometrical are not necessarily natural methods of 

 demonstration ; but that there is a real distinction cannot, I 

 think, be questioned. If it were not so, if the Student felt 

 that by studying a subject geometrically he acquired more real 

 insight into it than he could else have got, geometry would be 

 more popular in the University than it now is. In truth, the 

 difficulty of remembering many geometrical demonstrations is 

 in itself a proof of their artificial character ; for that which the 

 mind has once completely grasped it does not easily forget. 

 What we want is the introduction of a freer and more liberal 

 method*, and especially the abandonment of the notion that 

 anything is gained by a rigorous separation of geometry and 

 analysis. It is this which for the most part makes our geometry 

 pedantic, and many of our analytical text books dry and sterile. 

 If it be asked how such a change could be brought about, I am 

 inclined to think it could only result from a change in the 

 opinions of those on whom the character of our studies chiefly 

 depends, the Professors, Tutors, and Examiners. It could 

 hardly be made the subject of direct legislation. 



IV. The same remark would apply to the subject more 

 especially suggested by the seventh query, namely, the proper 

 limits of an undergraduate course of mathematics. In every 

 branch of mathematics there are parts which from their abstruse- 

 ness ought not to be introduced into the degree examination ; 



* What may be called the new geometry seems to be little studied in the Uni- 

 versity ; yet the method of which it makes so much use, namely, the generation 

 and transformation of figures by ideal motion, is more natural and philosophical 

 than the (so to speak) rigid geometry to which our attention has been confined. It 

 has been well said that the differential calculus is the symbolical expression of the 

 law of continuity, and probably the principles of the calculus would be better 

 understood if notions connected with this law were introduced at an earlier period 

 of our course. See, on the impossibility of severing our conceptions of space from 

 those of time and motion, Trendelenburg's Logische Untersuchungen. 



