302 Remarks on Professor Wallace’s Reply to B. 
plication of the proposed series fa, fb, &c. leads to resulis 
which have not been logically established, “ 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 altimate ratios, the expansion of bino- 
mials, multinomia's, &c.—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 bis 
method over that of every other one known, is wholly desti- 
tute of foundation It is somewhat amusing to observe that 
while 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 occurs in Vol. VII. page 284, in find- 
ing the series which expresses the log. (1+-r). 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 (1+ «)” 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 
md(it+a) m*.d?(1+x x a ae 
a 2 A &c.=m. prm(m =Dypst &c. 
This is true for all values of m. But if we put m=9, 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 U(1-+«) is deduced. 
Now though the true value is obtained, it is done by com- 
pletely shifting the hypothesis, according to Berkeley’s idea, 
from m finite, in which the development is possible, to m=9, 
