322 THIRD REPORT — 1833. 



equations. It has, in fact, been too much the custom of analysts 

 to consider the theory of numbers as altogether separated from 

 that of ordinary algebra. The methods employed have generally 

 been confined to the specific problem under consideration, and 

 have been altogether incapable of application vi^hen the known 

 quantities employed were expressed by general symbols and not 

 by specific numbers. It is to this cause that we may chiefly attri- 

 bute the want of continuity in the methods of investigation 

 which have been pursued, and the great confusion which has 

 been occasioned by the multiplication of insulated facts and 

 propositions which were not referable to, nor deducible from, 

 any general and comprehensive theory. 



Libri, in his Teoria dei Nmneri, and in his Memoir es de 

 MatMmatique et de Physique, has not merely extended the 

 views of Poinsot, but has endeavoured to comprehend all those 

 conditions in the theory of numbers, by means of algebraical or 

 ti-anscendental equations, which were previously understood 

 merely, and not symbohcally expressed. He has shown that 

 problems which have been usually considered as indeter- 

 minate are really more than determinate, and he has thus been 

 enabled to explain many anomalies which had formerly embar- 

 rassed analysts, by showing the necessary existence of an equa- 

 tion of condition, which ^must be satisfied, in order that the 

 problem required to be solved may be possible. By the aid of 

 such principles the solutions of indeterminate equations, at 

 least within finite limits, may be found directly, and without 

 the necessity of resorting to merely tentative processes. 



A great multitude of new and interesting conclusions result 

 from such views of the theory of numbers ; but the limits and 

 object of this Report will not allow me to discuss them in de- 

 tail, or to point out their connexion with the general theory of 

 equations, and with the properties of circular and other func- 

 tions. The reader, however, will find, in the second of the 

 memoirs of Libri above referred to, a general sketch of the 

 nature and consequences of these researches, which is unfor- 

 tunately, however, too rapid and too imperfectly developed to 

 put him in full and satisfactory possession of all the bases of 

 this most important theory. 



On the Solution of Numerical Equations. — The resolution 

 of numerical equations formed the subject of a truly classical 

 work by Lagrange, in which this problem, one of the most im- 

 portant in algebra, is not only completely solved, but the imper- 

 fections of all the methods which had been proposed for this 

 purpose by other authors are pointed out with that singular 

 distinctness and elegance which always distinguish his reviews 



