of the Binomial Calculus. 445 



it had been for so long a time enslaved. — The manuscript 

 containing this result was sent to Dr. Brinkley, to be re- 

 vised, and published in the Transactions of the Irish Aca- 

 demy. But the manuscript being in a very disordered state, 

 in consequence of my health having suffered from long and 

 intense application, it did not meet the countenance of Dr. 

 Brinkley. I sent notice, immediately after the refusal of 

 Dr. Brinkley, to the French Institute; and it would appear, 

 that this Institute thought as much about it as the Irish In- 

 stitute. The authority of the Royal Academy of Sciences of 

 Paris is no doubt great, but I am of opinion that the autho- 

 rity of demonstrated truth is infinitely greater. Truth, when 

 opposed to long established prejudices, has always to encoun- 

 ter the most serious obstacles. It is a bold innovation that 

 has demonstrated the absurdity of the principle of reasoning 

 employed in the Mccanique Celeste ; and the binomial calculus 

 has accomplished this. And it appears to me too, that the 

 physical hypothesis employed in that work is also absurd. 



When any arbitrary increase is given to the independent 

 variables, I confine the term dinomiation to the development, 

 arranging according to the arbitrary quantities added. I call 

 the leading term the finomial ; the second, the dinomial ; the 

 third, the second dinomial, &c. I give the term binomiation 

 to the development when no increase is given to the variables. 



Dinomial Theorem. — In the binomiation, the sign of any 

 term is the sign of this term combined with all those that fol- 



lowit ' Let( < r+A)" = x"(l-h?>) r - 



As x n is evidently common to all the terms in the binomia- 

 tion of (l+p) r , the theorem does not depend on any deter- 

 minate magnitude of X. Then p is not a determinate re- 

 lation. It will be sufficient, therefore, to show that the theorem 

 exists in the binomiation of ( + 1 + 1)", and of ( + 1—1)", n 

 being any relation whatever. With respect to (1+1)", when 

 n is any positive whole number, the theorem is evident. When 

 n is any complex number, the terms are at length alternately 

 positive and negative : but before this point the terms con- 

 verge; and after the commencement of convergence, every 

 term is greater than that which next follows it. Consequently 

 the sign of any term is the sign of this term combined with 

 all those that follow it. 



Considering now (1 — 1)", and binomiating, I get 



. n(n — 1) n 



i-« + -\-^- -&o. = o. 



Taking any term whatever of this series, and adding all the 

 negative terms that precede it to both sides, and subtracting 



all 



