i8o HERMANN VON HELMHOLTZ 



investigation extends to the sixth or eighth partial tone. By 

 establishing this last theorem, that difference of phase does 

 not come into the question, Helmholtz confirmed his previous 

 assumption that our sensation of different qualities of tone 

 is reduced to the fact that other nerve-fibres, corresponding 

 with the partials, are simultaneously excited along with the 

 fibres that respond to the fundamental tone. This simple 

 explanation would not suffice, if the difference in phase of the 

 deeper harmonics had to be considered. 



Helmholtz gave an enlarged account of his work in the 

 following year to the Nat. Hist. Med. Verein, at Heidel- 

 berg, in a lecture ' On Timbre '. He removed the restriction 

 that the vowel-sounds should be sung upon a single note (that 

 of a man's voice at B), and investigated all pitches of sung 

 vowels, finding that certain vowels are characterized by still 

 higher over-tones. 



In the paper laid before the Bavarian Academy, Helmholtz 

 refers to the great work, which he had termed a preliminary 

 study, published that year in the Reine u. Angew. Mathematik 

 'The Theory of Aerial Vibrations in Tubes with Open 

 Ends/ the contents of which he had already communicated 

 to the above Society at Heidelberg on March 15. This research, 

 with that mentioned above on Vortex Motion, must be reckoned 

 among the most brilliant of Helmholtz's mathematical achieve- 

 ments, only rivalled, and perhaps surpassed, by the work of the 

 last ten years of his life. 



' In 1891,' he writes, ' I have been able to solve a few 

 problems in mathematics and physics, including some that 

 the great mathematicians had puzzled over in vain from 

 Euler onwards : e. g. the question of vortex motion, and 

 the discontinuity of motions in fluids, that of the motions of 

 sound at the open ends of organ pipes, &c. But any pride 

 I might have felt in my conclusions was perceptibly lessened 

 by the fact that I knew that the solution of these problems 

 had almost always come to me as the gradual generalization of 

 favourable examples, by a series of fortunate conjectures, after 

 many errors. I am fain to compare myself with a wanderer 

 on the mountains, who, not knowing the path, climbs slowly 

 and painfully upwards, and often has to retrace his steps 

 because he can go no farther then, whether by taking thought 



