-302 Remarks on Professor Wallace'' s Reply to B. 



plication of the proposed series fa, fb, &c. leads to results 

 which have not been logically estabhshed, " eitherby Newton 

 or Leibnitz or any of their followers down to La Grange. 

 The whole of their methods, notwithstanding the application 

 of the principle of exhaustions, of indivisibles, of the theory 

 of limits, of prime and ultimate ratios, the expansion of bino- 

 mials, multinomials, &ic. — are still liable to the objections of 

 Berkeley, their reasoning being more or less infected with the 

 fallacia suppositionis, or, as he calls it, the shifting of the hy- 

 pothesis. — The results deduced from the single multiplica- 

 tion of Stainville's series are not liable to the objections of 

 the. fallacia suppositionis .'''' 



Upon the preceding extract, it may be remarked, that 

 having shown that Euler's demonstration is identical with 

 that published by Professor Wallace, it follows that no objec- 

 tion can be made to the one that does not apply with equal 

 force to the other, therefore the assertion of Professor Wal- 

 lace, of the superior excellency and logical precision of his 

 method over that of every other one known, is wholly desti- 

 tute of foundation. It is somewhat amusing to observe that 

 w^hile Professor Wallace is boasting of tbe great logical ac- 

 curacy of his deductions from this method, he stumbles upon 

 as complete a fallacia suppositionis as ever Dr. Berkeley 

 found fault with. This occui-s in Vol. VII, page 284, in find- 

 ing the series which expresses the log. (14-^)- This series, 

 though correct, being investigated by a process in which the 

 hypothesis is completely shifted. It is done by putting the 

 two developments of (l-f-x)"" page 284 VoL VII. equal to 

 each other, and neglecting the first term 1 of each series, 

 which mutually destroy each other, by which he obtains 



7n.l{l-^x) m^J^ (l-i-x) X ^ ^^ t 

 J -f Y^ 4- 8£C.=m. Y-i-m(m-])^-^+&;c. 



This is true for all values of m. But if we put w=0, it be- 

 comes simply 0=0, and in the present form determines no- 

 thing. Professor Wallace, however, divides the whole ex- 

 pression by m, which Berkeley contends ought not to be done 

 except when m is a real quantity ; after the division is per- 

 formed m is put=0, and the value of /(l-j-^-) 's deduced. 

 Now though the true value is obtained,- it is done by com- 

 pletely shifting the hypothesis, according to Berkelej's idea, 

 from m finite, in which the development is possible, to m=0, 



