52i FOURTH RKPORT — 1834. 



perplexed mathematicians. In particular, he professed to elu- 

 cidate the subject of the logarithms of negative and imaginary 

 quantities, which, at different periods, had occasioned contro- 

 versies between Leibnitz and Jean Bernoulli, Euler andD'Alem- 

 bert. 



The researches of others have since confirmed the views of 

 the author, whose claim to independent discovery and priority 

 of printed publication is undisputed. In a paper of subse- 

 quent date, published in the same volume of the Phil. Trans., 

 the Rev. John Warren of Cambridge, by original investiga- 

 tion, arrived at some of Mr. Graves's results. In June, 1832, 

 M. Vincent published at Lille, results identical in effect with the 

 author's principal formulae. M.Vincent claims to have antici- 

 pated Mr. Graves in their discovery, and appeals, in corrobora- 

 tion of this statement, to unpublished documentary evidence in 

 the archives of the Societe Philomatique, containing the Rapport 

 of MM. Ampere and Bourdon on a Memoire read August 18, 

 1827, as appears by the proces-verbal of that day. This Me- 

 moire is said by M. Vincent to have been substantially the 

 same as that of June, 1832, and to have been communicated to 

 M. Gergonne as early as April, 1825. Finally, Professor Ha- 

 milton, of Dublin, has deduced from his ingenious " Theory of 

 Conjugate Functions or Algebraic Couples" a complete confir- 

 mation of the author's system. 



Mr. Peacock, in his " Review of the recent progress of Analy- 

 sis," (page 267 of the Transactions of the Association for 1833,) 

 noticed the researches of Mr. Graves, but did not acquiesce in 

 his conclusions, which he conceived to be difficult to reconcile 

 with received opinions, and to be founded on the untenable 

 assumption of a periodic logarithmic base. It was for the pur- 

 pose of removing the impression which the high authority of 

 Mr. Peacock is calculated to produce that the Author presented 

 to the Association a second paper on the subject, in order to 

 invite the attention of analysts to a condensed statement of his 

 reasoning and results, exhibited in a more systematic and po- 

 pular shape than in his former essay. 



He is of opinion that the embarrassments and absurdities 

 which still encumber the doctrine of exponential functions have 

 chiefly arisen from calculating without fixed original principles ; 

 from occasionally regarding disintegrated properties, of partial 

 and collateral application to such functions, as the foundations 

 of essential and unlimited theorems ; from incautiously em- 

 ploying developments in unterminated series, without reference 

 to their complements and the limits of their accuracy ; and, 

 above all, from applying algebraic rules, that are a))propriate 



