316 THIRD REPORT — 1833. 



of the conditions of the problem itself. But even if we should 

 assume the impossibility of the problem, to the full extent of 

 Abel's demonstration, it is still possible that there may exist 

 solutions by means of undiscovered transcendents. It is, in fact, 

 quite impossible to attempt to limit the resources of analysis, or 

 to demonstrate the nonexistence of symbolical forms which may 

 be competent to fulfil every condition which the solution of this 

 problem may require. In conformity with such views, we may 

 consider the numerical roots of equations as the only discover- 

 able values of such transcendental functions ; but it is quite 

 obvious that such values will in no respect assist us in deter- 

 mining their nature or symbolical form, in the absence of any 

 knowledge of the course of successive operations upon all the 

 coefficients of the equation which were required for their de- 

 termination. 



Though we may venture to despair, at least in the present 

 limited state of our knowledge of transcendental functions, of 

 ever effecting the general resolution of equations, in the large 

 sense in which that problem is commonly proposed and under- 

 stood, yet there are large classes of equations of all orders 

 which admit of perfect algebraical solution. The principal pro- 

 perties of the roots of the binomial equation ^"—1=0, had 

 long been ascertained by the researches of Waring and La- 

 grange, and its general transcendental solution had been com- 

 pletely effected. Its algebraical solution, howevei*, had been 

 limited to values of n not exceeding 10 ; and though Vander- 

 monde in some very remarkable researches *, which were con- 

 temporary with those of Lagrange, had given the solution of 

 the equation x" — 1 = 0, as a consequence of his general me- 

 thod for the solution of equations, and had asserted that it 

 could be extended to those of higher dimensions, yet his solu- 

 tion contained no developement of the peculiar theory of such 

 binomial equations, and was so little understood, that his dis- 

 covery, if such it may be termed, remained al^arren fact, which 

 in no way contributed to the advancement of our analytical 

 knowledge. 



The appearance of the Disquisitiones Arithmeticee of the 



* Memoires de I'Academie de Paris for 1771. The result only of this solu- 

 tion was given, the steps of the process by which it was obtained being omitted. 

 This result has been verified by Lagrange in Note xiv. to his Traiie szir la 

 Resolution des Equations Nameriqties. Poinsot, in a memoir on the solution 

 of the congruence x" — 1 := M {p), which will be noticed in the text, has at- 

 tempted to set up a prior claim in favour of Vandermonde for Gauss's memo- 

 rable discovery ; in doing so, however, he appears to have been more influ- 

 enced by his national predilections in favour of his countrymen, than by a 

 strict regard to historical truth and justice. 



