equations of all orders, by continuous approximation. 335 
Addendum I. ( Vide Art. 21.) Note. But in this case, it will 
be more elegant to find the differences at once by the theorem 
A ,+ A'D m+ . 2±.*A'D m+2 <pR • &c. 
which, supposing r to be constant, is a sufficiently obvious 
corollary to the theorem in Art. 7. All the results may then 
be derived from the first column by addition. Thus, for the 
latter transformations in Ex. II. Art. 22, the preparatory 
operation would be 
1st. Terms. 
Diff. 1st. 
2nd. 
3rd. 
1000 
— 369 
66 
6 
— 4cc 
63 
6 
3° 
3 
1 
and the succeeding terms would be found by adding these 
differences in the usual way to the respective first terms. 
Addendum II. It is with pleasure that I refer to the Impe- 
rial Encyclopaedia (Art. Arithmetic) for an improved method 
of extracting the cube root, which should have been noticed 
in the proper place, had I been aware of its existence ; but it 
was pointed out to me, for the first time, by the discoverer, 
Mr. Exley, of Bristol, after this Essay was completed. It 
agrees in substance with the method deduced in Art. 25, 
from my general principle, and affords an additional illustra- 
tion of the affinity between that principle and the most im- 
proved processes of common arithmetic. 
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