374 GUSTAV ROBERT KIRCHHOFF. 



not necessary to give a complete definition of it. But the imperfection 

 of this definition introduces no obscurity into results ; for the introduc- 

 tion of forces in this way only serves to simplify the modes of state- 

 ment, and to express briefly equations which without the help of this 

 name would be clumsily described by words. It is sufficient for remov- 

 ing all obscurity, to give so wide a definition to forces that every law 

 of mechanics in which forces are named can be expressed by equations : 

 and this happens in a striking manner. 



Of KirchhofF and his lectures Robert von Helmholtz writes thus, as 

 translated in the Popular Science Monthly : " Plis favorite work, and 

 the one having the most enduring results, was his lectures on mathe- 

 matical physics. His address was impressive by reason of the elegance 

 and precision of his statement. Not a word was wanting, not a word 

 was in excess ; never an error, an obscurity, or an ambiguity. Re- 

 markable also was the exactness of his calculations, — a matter of 

 extreme difficulty to laymen. The whole material arranged itself 

 before the eyes of the class in the form of a nicely adjusted master- 

 work of scientific art, so that every part exerted its full effect on the 

 others, and to witness one of his deductions was a real esthetic enjoy- 

 ment. The complete understanding of his reasoning on these most 

 difficult subjects implied, of course, some knowledge of the mathemati- 

 cal language which was his vehicle of thought ; and it might happen, 

 and did in fact sometimes happen, that a hearer could not comprehend 

 why KirchhofF made this particular deduction and not some other; but 

 every one was able to follow his course of thought, consider it, and 

 render it correctly. So that, paradoxical as it may appear, it was 

 not impossible, without having really understood Kirchhoff, to repro- 

 duce his lectures from the notes into a respectable book. Kirchhoff 

 was able to give his lectures uninterruptedly in Berlin for nine years. 

 But we who heard him could remark the effort they caused him, and 

 how he had to husband his strength. Yet he was always punctual, 

 and the quality of his teachings was never depreciated. Finally, in 

 1884, the doctors forbade him to read ; and although he was able to 

 resume this his favorite occupation for a time, it was evident that his 

 nervous system was shattered." 



Kirchhoff was about fifty years old when he was called to Berlin. 

 He had already done his greatest work on the spectrum, and published 

 it. But his eyes had suffered from an accidental exposure to the sun, 

 and his foot had been seriously injured in a way which impaired his 

 general health. After 1882, the date of his Abhandlungcn, he pub- 

 lished a few papers. 



