THE SIX GATEWA YS OF KNOWLEDGE. 277 



difference in singing, and of all the expression 

 put into a note or a sequence of notes in sing- 

 ing, that cannot be written down. There is, on 

 the written or printed page, a little wedge showing 

 a diminuendo, and a wedge turned the other 

 way showing a crescendo, and that is all that 

 the musician can put on paper to mark the 

 difference of expression which is to be given. 

 Well now, all that can be represented by a whole 

 page or two pages of orchestral score, as the 

 specification of the sound to be produced in, say, 

 ten seconds of time, is shown to the eye with 

 perfect clearness by a single curve on a riband 

 of paper a hundred inches long. That to my 

 mind is a wonderful proof of the potency of 

 mathematics. Do not let any student in this 

 Institute be deterred for a moment from the pur- 

 suit of mathematical studies by thinking that the 

 great mathematicians get into the realm of four 

 dimensions where you cannot follow them. Take 

 what Professor Cayley, himself, in his admirable 

 address which I have already referred to, told 

 us of the beautiful and splendid power of mathe- 



