was received from Mr. Jacob Cist of Wilkesbarre, describing 

 a singular lusus naturce. This paper, though valuable, viras 

 considered as not coming within the general plan of the Aca- 

 demy's journal ; but it has since been published in another 

 journal, whose design was more comprehensive. 

 [To be continued.] 



LVII. Reply to Mr.DAViEs's '^Further Thoughts on iWr.HE- 

 rapath's Demonstration." By P. Q. 



TT is really difficult to say which is the most striking in 

 -* Mr. Davies's criticism — the skill and ingenuity of his at- 

 tack, or the honourable candour and liberality which he dis- 

 plays towards the party opposed. We may with propriety 

 observe, that " intaminatis JFulget honoribus." 



If in the observation, " whether Mr. Herapath has failed 

 in his usual precision of expression," Mr. Davies means to 

 imply that Mr. H. has written with a little too much brevity 

 for perfect elementary perspicuity, he is probably not far from 

 the truth. But this ought not to be charged as a fault so much 

 on Mr. Herapath as on the limits to which he was of neces- 

 sity confined. It would have been impossible to comprise so 

 many points, as he has in that paper treated of, within any 

 reasonable bounds, had he not in every part studied the ut- 

 most compression. 



As Mr. Davies has transferred his attack from the pre- 

 viously contested points to others, I shall in my defence of 

 Mr. H. follow his example, and try if I cannot convince 

 that gentleman, without " any new species of ' mathematical 

 magic,' " that Mr. Herapath is here too equally as invulnera- 

 ble as in the positions abandoned. Mr. H. assumes 



/• + u = «, any integer whatever, 

 and tells us that " r or v may be any number rational, irrar 

 tional, or imaginary " a fact, the truth of which is so obvious 

 that any one would smile at an attempt to establish it by de- 

 monstration. Mr. Herapath then adds : " and since the sum 

 n of these numbers " (r and v) " is an indeterminate positive in- 

 teger, they mil in j)oint of value he independent" That this, 

 which is one of the disputed points, is correct, may be shown 

 in an instant. For suppose one of the quantities, r for in- 

 stance, at the time it has any non-integral value to remain 

 constant, whilst n varies any how through integral values only; 

 then A w is evidently = A n, 



V still retaining its non-integral value. That is, v is a non- 

 integer and a variable, whose variations are not necessarily at 



aU 



I 



