equations of all orders , by continuous approximation. 327 
Consequently, 
1 116143772) i72i4582i897s( 1542326590,22 
This third correction is carried two places beyond the extent 
of the divisor, for the sake of ascertaining rigidly the degree 
of accuracy now attained. For this purpose, we proceed thus: 
628 &c. x 154 &c. =, 968, &c. is the true correction of the 
last divisor. Our anticipated correction was i,ooo. For 
which if we substitute 968 &c. it will appear that our divisor 
should have ended in 1,678, &c. instead of 2. The error is, 
,322 &c. which induces an ultimate error of (111 &c. : I54 
&c. : 1,322 &c. &c. :),44 &c. Consequently, our third cor- 
rection should be . . . . 1542326590,66, &c. agreeing to 10 
figures with the value previously determined. And the root is 
x= 2.094551481542326590, &c. 
correct in the 18th decimal place at three approximations. 
So rapid an advance is to be expected only under very 
favorable data. Yet this example clearly affixes to the new 
method, a character of unusual boldness and certainty; ad- 
vantages derived from the overt manner of conducting the 
work, which thus contains its own proof. 
The abbreviations used in the close of this example, are of 
a description sufficiently obvious and inartificial ; but in order 
to perfect the algorithm of our method in its application to 
higher equations, and to the progress by simple digits, atten~ 
tion must be given to the following general principles of 
Compendious Operation. 
27. We have seen that every new digit of the root occa- 
sions the resolvend to be extended n figures to the right, 
and the m th derivee n — m figures ; so that if the work be car- 
ried on as with a view to unlimited progress, every new 
