THE 

 LONDON, EDINBURGH and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



[THIRD SERIES.] 



DECEMBER 1844. 



LXIX. Observations on Mr. Graves's Theory of Imaginary 

 Logarithms. By J. R. Young, Professor of Mathetnatics 

 in Belfast College *. 



T^HE extended and accurate theory of imaginary loga- 

 -■- rithms, first propounded by Mr. Graves in tlie Philo- 

 sophical Transactions for 1829^tj is pretty well known to the 

 readers of this Journal, its merits having been very fully dis- 

 cussed by Mr. Graves himself in the volumes of the Philoso- 

 phical Magazine for 1836 (S. 3. vols. viii. and ix.), in reply to 

 certain objections that had been advanced against it. In these 

 latter explanations of Mr. Graves, qualifications and conces- 

 sions appear to have been made which I think were unneces- 

 sary, and which seem to me to deduct very considerably from 

 the importance of his discovery. I cannot but consider that 

 these modifications of the author have tended somewhat to 

 retard the general reception of his improved theory ; they cer- 

 tainly operated unfavourably on my own mind, and for a long 

 time led me to entertain the conclusion, that what Mr. Graves 

 had done was not a generalization of the previously existing 

 theory, as it was left by Euler, but rather, to use his own 

 words, " the erection of a wing" to the old building, which 

 he seemed to regard as in itself complete, and as much as the 

 original foundation could sustain. 



In this brief communication it is my object to show that 

 Mr. Graves's extended theory is necessary to complete the 

 fabric which Euler had left unfinished ; that it occupies no 

 new ground, nor requires any amplification of the principles 

 universally admitted before the publication of Mr. Graves's 

 researches. 



Let the arithmetical value of the Neperian logarithm of the 

 real number a be represented by la, and the general loga- 

 • Communicated by the Author. f See Phil. Mag. S. 2. vol. v. p. 378. 



Fhil, Mag. S. 3. Vol. 25. No. 168. Dec. 1 844. 2 D 



