332 Mr. John Walsh on the Calculus of Variations. 



preceding extracts refers. The Institute of France calls it a 

 pretended demonstration. Then all our numerical divisions 

 are only pretended. I have given the only general demon- 

 stration of the formula of the binomial that has as yet ap- 

 peared. It was read by the Royal Society of London, June 

 6, 1821. Had I sent it to the Institute of France, it would 

 have been called pretended, and rejected. The second ex- 

 tract refers to the dinomial theorem. The report asserts, 

 that the theorem evidently supposes, the second term of the 

 binomial to be less than the first. The paper of mine in the 

 Philosophical Magazine for last June, demonstrates that the 

 Institute is in error in coming to such a conclusion; for the 

 theorem is demonstrated in that paper when both the terms 

 of the binomial are equal to each other. The dinomial 

 theorem is a general law of all series. It has banished more 

 errors from algebra than has the system of Copernicus ba- 

 nished from physical astronomy. The following is its enun- 

 ciation in the most general sense. 



In every series, the sign of any term after the fast, is the 

 sign of that term combined with all those that follow it. 



This grand theorem has banished fluxions, differentials, 

 and the calculus of variations, and has extended the domain 

 of algebra without limit. 



The following is from the last report which the Academy 

 of Sciences has given with respect to the binomial calculus, 

 that I have received : 



" Les commissaires ont ete d'avis que les memoires de 

 M. Walsh n'apprenoient rien de nouveau, ils croient ne de- 

 voir attacher aucune importance au nom que l'auteur voudra 

 donner a son calcul, et se dispenseront de fixer plus long-temps 

 sur ces objets 1' attention de 1' Academic" 



" (Signe) Poisson, Cauchy, Rapporteur." 

 ■ Lundi, lfiJuin, 1823."- 



The influence which the infinitesimal calculus has so long 

 exercised in geometry and analysis, has not permitted the 

 Royal Academy of Sciences of Paris to perceive clearly its 

 way through the binomial calculus. The preceding demon- 

 stration of the fifth proposition of the second book of Euclid, 

 first, demonstrates the absurdity of the logic of the infinitesimal 

 calculus ; secondly, it developes the general theory of maxima 

 and minima; and, thirdly, it demonstrates the nonentity of 

 the calculus of variations. 



Cork, Sept. 30, 1824. 



LVII. On 



