1819.] Royal Academy of Sciences. 223 



the final result will be as exact, and the demonstration as well 

 established, as if these irrational values had not been employed. 



From the very simple analysis of the author, it also follows, 

 that if the general formula of the roots of unity does not contain 

 radicals, the exponents of which will not divide (p — 1), then 

 there is not, in the whole expression, a single radical which is 

 not related to an exact power of the same degree, or rather to a 

 residual of that power ; so that, by the restitution of certain 

 multiples of p to these residuals, the expression will become, in 

 all its parts, commensurable and entire, and will not show in any 

 part any sign of an impossible operation. 



But if there are found in the formula roots which are not 

 divisors of (p — 1), there will be in it, under these radicals, 

 numbers which will not be residuals of powers of the same 

 degree, and consequently the formula will contain irrational 

 quantities which can never be reduced to entire numbers in 

 respect to p. But these irrational quantities may be exact 

 powers of irrationals of the same form, so that the radical opera- 

 tion that is pointed out can be executed ; and then, in the addi- 

 tion of these similar values, the incommensurables are destroyed 

 of themselves, and the formula will always lead with equal 

 precision to the entire roots of the proposition, when this equa- 

 tion will admit such roots. 



These are the principal points of this remarkable theorem, 

 which offers, as the author observes, the first and only example 

 of the application of algebra to the theory of numbers. 



M. Poinsot has examined this theorem still more deeply, and 

 explained it by a multitude of examples, which exhibit a number 

 of curious theorems concerning the residuals of the powers of 

 the superior degrees. He also applies it to the determination 

 of the primitive roots of prime numbers ; and lastly, he extracts 

 from it some general truths in respect to algebra, which were 

 apparently impossible to be discovered by any other method. 



As to the rest, this theorem of the binomial equations extends 

 to any equation, the algebraic resolution of which shall be 

 esteemed as known. The author points out, in a few words, 

 this general demonstration at the commencement of his memoir; 

 " but I wished," he says, "to study more particularly the binomial 

 equations, because they are, as it were, the key of all the others, 

 because they alone are able to show the intimate nature of roots : 

 those remarkable signs, which exhibitthe essence of algebra by that 

 equivocation of the different senses in which they may be taken, 

 and the employment of which, in the chain of mathematical 

 reasoning, exhibits the most precise difference between analysis 

 and synthesis." 



The author proposes to proceed with this curious approxima- 

 tion of algebra, and the theory of numbers, and to show that the 

 general principles of algebra, properly so called, have their origin 

 in the simple consideration of the order, or of the mutual dispo- 



