334? Mr. Horner's new method of solving numerical 
of these will reduce the process to the seventh order ; one- 
more reduces it to the sixth order ; two more, to the fifth, 
&c. The last 27 figures will be found by division alone. 
But if the first additional correction is taken to 8 figures, 
and the second to 16, on the principle of Example II, we pass 
from the 8th order to the 4th at once, and thence to the 1st 
or mere division, which will give the remaining 29 figures. 
This mode appears in description to possess the greater sim- 
plicity, but is perhaps the more laborious. 
It cannot fail to be observed, that in all these examples a 
great proportion of the whole labour of solution is expended 
on the comparatively small portion of the; root, which is con- 
nected with the leading process. The toil attending this part 
of the solution, in examples similar in kind to the last, is very 
considerable ; since every derivee is at this stage to receive 
its utmost digital extent. To obviate an unjust prejudice, I 
must therefore invite the reader's candid attention to the fol- 
lowing particulars : 
In all other methods the difficulty increases with the extent 
of the root, nearly through the whole work ; in ours, it is in 
a great measure surmounted at the first step : in most others, 
there is a periodical recurrence to first conditions, under cir- 
cumstances of accumulating inconvenience ; in the new me- 
thod, the given conditions affect the first derivees alone, and 
the remaining process is arithmetically direct , and increasingly 
easy to the end. 
The question of practical facility may be decided by a very 
simple criterion ; by comparing the times of calculation which 
I have specified, with a similar datum by Dr. Halley in favor 
of his own favorite method of approximation. (Philosophical 
Transactions for 1694.) 
