REPORT ON THE RECENT PROGRESS OF 

 ANALYSIS (THEORY OF THE COMPARI- 

 SON OF TRANSCENDENTALS)*. 



1. THE province of analysis, to which the theory of elliptic 

 functions belongs, has within the last twenty years assumed a 

 new aspect. A great deal has doubtless been effected in other 

 subjects, but in no other I think has our knowledge advanced 

 so far beyond the limits to which it was not long since con- 

 fined. 



This circumstance would give a particular interest to a his- 

 tory of the recent progress of the subject, even did it now 

 appear to have reached its full development. But on the con- 

 trary, there is now more hope of further progress than at the 

 commencement of the period of which I have been speaking. 

 When, in 1827, Legendre produced the first two volumes of his 

 1 The*orie des Fonctions Elliptiques,' he had been engaged on 

 the subject for about forty years ; he had reduced it to a sys- 

 tematic form ; and had with great labour constructed tables to 

 facilitate numerical applications of his results. But little more, 

 as it seemed, was yet to be done ; nor does the remark of 

 Bacon, that knowledge, after it has been systematized, is less 

 likely to increase than before, seem less applicable to mathe- 

 matical than to natural science. Nevertheless, almost immedi- 

 ately after the publication of Legendre's work, the earlier re- 

 searches of Abel and Jacobi became known, and it was at once 

 seen that what had been already accomplished formed but a part, 

 and not a large one, of the whole subject. 



To say this is not to derogate from the merit of Legendre. 

 He created the theory of elliptic functions ; and it is impossible 



* Report of the Sixteenth Meeting of the British Association; held at Southamp- 

 ton, in September, 1846. 



