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XXV. Reply to Mr. Davies's Critique on Mr. Herapath, 

 By A Correspondent. 



To the Editor of the Philosophical Magazine and Journal. 

 Sir, 

 IV/TR. DAVIES's remarks in your last Number on some of 

 ^^ Mr. Herapath's mathematical writings are tinctured 

 with so much good feeling, that I cannot believe the incorrect 

 views he has given of Mr. H.'s demonstration of the binomial 

 theorem could be occasioned by any thing but haste or mis- 

 conception. It is however a fact, that his statement of Mr. 

 Herapath's demonstration is by no means just; and as all his 

 other observations are founded on this, I trust you will allow 

 me a small space to do Mr. H.'s labours that justice they de- 

 serve. 



After slightly commenting on Mr. Herapath's proof for po- 

 sitive integers, which is too much of the axiomatic kind to need 

 a word in its defence, Mr. Davies says (Phil. Mag. for August 

 1825, p. 117): "Here, however, Mr. Herapath is obliged 

 to pause, and to call in the aid of a new principle on which 

 to found his inferences ; viz. that because it 'was true 'when r and 

 V 'were integers, it must be true 'when they are fractions, and 

 even 'when they are irrationals and imaginaries too .'" A little 

 lower Mr. D. adds : " The investigation lays it down as an 

 essential datum that r and v shall be integers; and immediately 

 proceeds on the supposition of their heing fractions, irrationals, 

 or imagitiaries, taken at pleasure ! " 



Now as soon as Mr. Herapath has demonstrated the bino- 

 mial theorem for a positive integral power, which he denotes 

 by n, the only exponent yet introduced, he proceeds to the 

 consideration of non-integral powers, (Phil. Mag. for May 

 1825, p. 323,) thus 



" Again for no7i-integral expotients. 



" Suppose the second, third, fourth, S^c. coefficients of the 

 rth power are represented by 2^, 3^, ^r>" &c. A little lower 

 Mr. H. says : " nmltiplying the rth po'wer by the vth, the actual 

 operation gives," &c. These are Mr. Herapath's introduc- 

 tions of the ?th and t»th indices; and certainly nothing is here 

 implied of their being integers. He is in fact distinctly speak- 

 ing of *' non-integral exponents," and at the very beginning 

 of the next page recalls our attention to the non-integral signi- 

 fication of?- and V, in the following terms : "No-w, ifn = r + v, 

 and n be as before an integer," (Mr. H., as I have said, had 

 previously employed n alone in his investigation for the inte- 



Z 2 ger 



