536] note on Lagrange’s demonstration of taylor’s theorem. 
495 
really is shown is that admitting such an operation to be possible in regard not only 
by the successive repetitions of this operation and by dividing by the proper numerical 
denominator. 
By what precedes, any objection in regard to convergency, I regard as irrelevant; 
and if it is said that the above-mentioned single assumption is not granted, I would 
either ask “ What is a function ”—or I would content myself with the hypothetical 
statement—if f(x) be such that f(x + h) is expansible ut supra, then Taylor’s theorem. 
In regard to the demonstration given by Mr Todhunter, it implicitly assumes that 
x and h are both real, and (although doubtless possible) it would be considerably 
more difficult to find an analogous demonstration of the formula involving f n+1 (x + Oh) 
in the case of x and h imaginary. But the formula with the term in question is not 
(nor does Mr Todhunter consider it as being) Taylor’s theorem; to obtain from' it Taylor’s 
theorem, we require (in the foregoing point of view) the property that h n+1 f(x 4- Oh) is 
expansible in a series involving h n+1 and the higher powers of h, that is, the very 
property that f(x + h) is expansible in positive powers of h. 
Moreover admitting that the formula with the term f n+1 (x + 0h) is demonstrable 
for imaginary values of x, li, the formula is meaningless in the case where x, h are 
one or both a symbol or symbols of operation: 0 would certainly have no definable 
numerical magnitude, and if it is considered as meaning anything, then the equation 
in question is a mere definition of what it does mean, and ceases to be a theorem 
in regard to f{x + li). It is impossible, in a quantitative algebra such as is presupposed 
in the method of limits, to put any meaning on the equation 
which however I regard as a legitimate particular form of Taylor’s theorem.