536] 
493 
536. 
NOTE ON LAGBANGE’S DEMONSTBATION OF TAYLOB’S 
THEOBEM. 
[From the Messenger of Mathematics, vol. I. (1872), pp. 22—24.] 
I take the occasion of the publication of the last edition of Mr Todhunter’s 
Treatise on the Differential Calculus to make some remarks on the demonstration in 
question. Mr Todhunter proposes to himself to exhibit a comprehensive view of the 
Differential Calculus on the method of Limits; but he very properly introduces in some 
cases demonstrations founded upon other views of the subject, pointing out that this 
is the case, ands explaining or indicating his objections. Thus (Chapter VI.) upon 
Taylor’s Theorem, he remarks “Before we offer a strict demonstration of the theorem 
in question, we shall notice the method which it was usual to adopt in treatises on 
the Differential Calculus not based on the doctrine of limits,” and then, after giving 
a demonstration depending on the relation J^/(# + /<.) = + ^XO) he S oes on 
“ There are numerous objections to the method of the preceding articles, and especially 
the use of an infinite series, without ascertaining that it is convergent, is inadmissible; 
we proceed then to a rigorous investigation,” which investigation (after Mr Homersham 
Cox) is a demonstration of the equation 
fix + h) =f(æ) + hf (op) 
h n 
h n+1 
(6 between 0 and 1) whence “if the function f n+1 (x + 6h) is such that by making n 
h,n+1 
.sufficiently great the term ^ ^yf l+1 (x + 6h) can be made as small as we please, then 
by carrying on the series 
/(«) + ¥' 0r) + *\f («) + j/" (*) + • • • 
1 This demonstration is similar in principle to Lagrange’s but I think his is preferable ; viz. the principle 
made use of by Lagrange is that the series has the same value whether x is changed into x + k, or h into h + lc.