116 Mr. T. S. Davies on the 



tliat <p {x + n/i) = :p X + 71 . A4>^ + "'j^"" . A' 4> .r + ... ; 

 and that = .r""'^ ;* and of a thousand other 



t n — r n — r ' 



dx A 



formuliB : indeed, of every possible formula that is true. 



I do by no means intend to assert that all these propositions 

 are equally clear and obvious to the untutored mind; but 

 merely that they are all equally the result of a generalization 

 of facts, which had been previously observed to be true in every 

 one of the numerous instances vk^hich had been examined. The 

 object of the generalization was to discover an empirical ex- 

 pression which should, mutatis mutandis, comprehend all the 

 possible separate cases; and we felt justified in the use of the 

 rule thus formed, so long as no exception to its generality was 

 discovered, whilst from every instance of its accordance with 

 separate cases that came under our notice we felt it invested 

 with an additional degree of certainty. This was asserted in 

 the binomial, theorem ; and by the analogy in integer num- 

 bers between that and some of the differential theorems, the 

 same generalization was suggested in this class of theorems 

 which had been successfully tried in the binomial. Such an 

 extension was very obvious ; but it required us to generalize 

 (if, indeed, that could be done) the signification of the index 

 of the characteristic, so as to include fractions, irrationals, and 

 imaginaries, in order to render the resulting expressions in- 

 telligible. This being done, it became easy to verify the 

 theorem by application to particular cases, and thus to place 

 it amongst the admitted principles of the science. 



My intention in the present paper, however, was principally 

 to notice Mr. Herapath's Demonstration of the Binomial 

 Theorem (Phil. Mag. May 1825), and his extension of it to 

 the development of any function in terms of the finite diffe- 

 rences of that quantity ; and to point out where that gentle- 

 man has deceived himself by an appearance of greater rigour 

 than he had actually obtained in his demonstration. I shall 

 also notice, en passant, that given by the Rev. Arthur Brown in 

 his " View of the First Principles of the Differential Calculus," 

 p. 59, Cambr. 1824'. 



In the first place, upon what authority but that of induction 

 does Mr. Herapath assume the nih line in his class of equa- 

 tions (B), p. 323? The induction, I grant, is easy, and the 

 law tolerably obvious : but it is induction still, and the princi- 

 ple of generalization is essentially involved. The formula has 

 been found to hold true in all the integer numbers that have 



* Mr. Herapath, Phil. Mag. May 1825, vol. Ixv. p. 327- 



been 



