equations of all orders, by continuous approximation. 325 
how much more concise it is than even the abbreviated statement 
of the old process. (See Hutton's Course.) 
The station of 1, 2, &c. numeral places respectively, which 
the closing addends occupy in advance of the preparatory 
ones, is an obvious consequence of combining the numeral 
relation of the successive root-figures with the potential rela- 
tion of the successive derivees. In fact, as is usual in arith- 
metic, we tacitly regard the last root-figure as units, and the 
new one as a decimal fraction; then the common rules of 
decimal addition and multiplication regulate the vertical ali- 
neation of the addends. 
2 6. The advantage of mental verification is common to the 
solution of equations of every order, provided the .<-uccessive 
corrections of the root be simple digits : for the parenthetic 
derivations will, in that case, consist of multiplying a given 
number by a digit, and adding the successive digital products 
to the corresponding digits of another given number ; all 
which may readily be done without writing a figure interme- 
diate to these given numbers and the combined result. For 
this reason the procedure by single digits appears generally 
preferable. 
Nevertheless, to assist the reader in forming his own 
option, and at the same time to institute a comparison with 
known methods on their own grounds, I introduce one ex- 
ample illustrative of the advantage which arises from the 
anticipatory correction of the divisors spoken of in Art. 24, 
when the object is to secure a high degree of convergency 
by as few approximations as possible. The example is that 
by which Newton elucidates his method. I premise as the 
depreciators of Newton do, that it is an extremely easy 
