272 Mr. E. H. M. Bosanquet on the History of the 



ance, the modem student cannot afford to dispense with 

 Smith's most powerful exposition. But it does not account 

 for our modern conception of beats, regarded as variations in 

 the intensity of certain notes. We have still to find out in 

 what way such variations of intensity arise out of the system 

 of recurrences Smith describes. 



With respect to the supposed difficulty of Smith's proposi- 

 tions, I am confident that any one who has mastered the above 

 example will find the difficulties gone, in all that part of the 

 work which is concerned with the doctrine of beats. 



Young's criticism on Smith's work is, that he achieved 

 nothing. De Morgan thinks that Young, and others whom 

 he quotes, did not understand Smith. The passage of Young 

 which De Morgan cites as a proof of this, however, is very 

 good sense, and undoubtedly represents the fact correctly on 

 Young's own theory. 



Whether Young understood Smith or not, and whether or 

 no Young's theory will now bear strict criticism, there can be 

 no doubt that Young's work had a most beneficial influence 

 on the progress of the subject. He is remarkably clear about 

 a number of important points ; and on some his exposition is 

 not surpassed at the present day. 



Young clearly pointed out the mode in which the ordinary 

 beats (which become Smith's short cycles in or near conso- 

 nances) are numerically related to the Tartini tones (combina- 

 tion-tones). He uses a phraseology which is now seen to be 

 objectionable, when he says that the beats become the Tartini 

 tones. This would involve, in our quantitative language, the 

 statement that the whole of the energy of the beats is trans- 

 formed into combination-tones, to which statement it is impos- 

 sible to assent. So that I differ from Young's exposition as 

 I differ from that of Konig, where he assumes the passage of 

 beats of great frequency into a note, and compares the beats 

 with impulses, as if they contained nothing else but the impulse. 

 But this difference of view will only touch the mode in which 

 the notes arise out of the beats. One might almost say that 

 it is a mere question of language ; but it is not so. The ele- 

 ment of quantitative transformation is essentially involved. 



The facts, however, are of the first importance ; and it is 

 not too much to say that Young's statements fully anticipate 

 the principal conclusions of Konig's great paper, and give the 

 facts completely and compendiously. It is no use attempting 

 a detailed analysis of Young's position, as it is substantially 

 the same as that of Konig. 



Of the other writers who occupy a similar position we need 

 only mention De Morgan, from whose analysis many have 



