Mr. T. «. Davies on Mr. Herapath's Demonstration. 21 r> 



itself the utmost degree of exactitude and certainty. Let us 

 then divest the science of its artificial connexion with the my- 

 sticism of innate ideas and universal truths cognisable by us 

 through other media than sensation and experience ! When 

 we can fairly understand the evidence upon which our know- 

 ledge actually rests, we shall have done much towards weeding 

 its details of many deformities and irrelevant operations, and 

 rendering not only the means of acquisition but those of dis- 

 covery more facile than the most sanguine of us can at pre- 

 sent form any notion of! 



Let us consider, then, — if indeed the error be not too ob- 

 vious to need remark at all, — the position laid down by Mr. 

 Herapath : w^hen 



>■ + !' = «= indetenninate intego; 

 that r and v " xvill in point of value he independent." I take 

 for granted here, that the distinctions in "value" here refer 

 to integer or fractional values of the symbols ; it is the oidy 

 interpretation of which I can perceive the application in the 

 present inquiry. Surely then r and v are mutually dependent: 

 if not, the equation ;• + u = integer is destroyed. Mr. Hera- 

 path's conditional equation and his conclusion cannot there- 

 fore simultaneously exist; and of course the reasoning which 

 is built upon that simultaneous existence must also be sub- 

 verted by this consideration. Indeed the " independence of the 

 functions" r and v is just of the same character as would be 

 the assumption of independence of an angle and its comple- 

 ment, or of a number and its reciprocal: and a demonstration 

 built upon assumptions like these would be equally as valid as 

 Mr. Herapath's demonstration of the binomial. 



Mr. Herapath proceeds — " And because in the two right 

 hand members of this" (an equation derived solely ^xom his de- 

 monstration for integer values and the principles of combina- 

 tions, in conjunction with n = 7+ w), " r and v are indepen- 

 dent variables, these members when duly reduced must not 

 contain any p)roduct of the powers of the variables; for if they 

 did, the function of either variable xcotdd be affected by the 

 changes of the other variable, which it should not." Will 

 not then a variation take place in any specific function of a 

 fraction and its complement, by changing the values of one of 

 these? It "is incumbent, at all events, on Mr. Herapath to 

 show the truth of such a jirinciple, before we can atlmit its 

 application here — a task which when accomplished will in- 

 troduce a new species of " mathematical magic," more refined, 

 and not less ludicrous, than that upon which Berkehjy and 

 Maseres exercised their castigating irony. 



M m 2 it 



