118 Mr. T. S. Davies on the 



ing upon periodical functions (Phil. Mag. November 1824, 

 p. 333) where the indices are fractional, we may m-ge the same 

 objections as we liave done against his reasoning upon the 

 binomial. Those assertions mai/ be true — and they probably 

 are true: but no proof of its truth has yet been given; and I 

 much mistake the nature of demonstration, if any proof but 

 that derived from successive inferences can ever be established. 

 The demonstration given by the Rev. Arthur Brown, to 

 which I before referred, proceeds in the usual manner when 

 71 is a positive integer. This integer he next supposes con- 

 verted into an equivalent vulgar fraction ( — -)j and argues 

 that it still is true. — Granted. But he proceeds, ^^ Since this is 

 true "when (-—) is a whole number, it will also be true 



■when ( — J is a fraction! (View, p. 60.) This is the 



work in which, after an imsparing review of the state of aca- 

 demic mathematics, the author has " endeavoured to give a 

 rigid demonstration of every proposition !" Pref. xxi. 



Some yeai's ago the views entertained by Mr. Herapath oc- 

 curred also to myself: though of the "more general calculus" 

 to which he refers (but of which he so carefully conceals all 

 traces) I am unable to form an opinion. My investigations 

 were conducted on the assumed truth of the differential for- 

 mulae when extended to any form whatever of 7i : and I had 

 examined most of the functions noticed by Mr. H., together 

 with many important ones to which he has not made any re- 

 ference, though doubtless they have more or less attracted his 

 attention. A concurrence of circumstances — amongst which 

 the annunciation of Mr. Herapath's discoveries more than 

 two years ago in the Phil. Mag. ; the difficulty of satisfying 

 myself at that time concerning the fundamental principles of 

 analytical certainty ; and the force of those claims upon my 

 attention, which I considered paramount even to the claims 

 of science; — these prevented me from pursuing the inquiry 

 so far as under more auspicious circumstances I probably 

 should have done : and from knowing that the question 

 was likely to be fully investigated by a mind of the vigorous 

 constitution of Mr. Herapath's, and with the aid of a new and 

 more extensive calculus, I felt altogether unanxious concerning 

 the state of my own investigations. It will be scarcely neces- 

 sary, then, to disclaim all hostile feelings in the remarks which 

 I have made ; and to assure that gentleman, that, but for the 

 error into which I conceive he has fallen, and the consequent 

 loss of much valuable time in vainly attempting impossibili- 

 ties, 



