2.—Then (by the Rule of Three),

As the sum of the given number, and double the assumed cube, is to the sum of the assumed cube, and double the given number, so is the root of the assumed cube, to the root required, nearly.

3.—Or as the first sum,
is to the difference of the given, and assumed cube,
so is the assumed root,
to the difference of the roots, nearly.

4.—Again, by using, in like manner, the cube of the root last found as a new assumed cube, another root will be obtained still nearer. Repeat this operation as often as necessary, using always the cube of the last-found root, for the assumed root.

Example.—To find the cube root of 21035·8.

By trials it will be found first, that the root lies between 20, and 30; and, secondly, between 27, and 28. Taking, therefore, 27, its cube is 19683, which will be the assumed cube. Then by No. 2 of the Rule

19683 21035·8
2 2
39366 42071·6
21035·819683·
As60401·8: 61754·6:: 27: 27·6047 the Root, nearly.
Again for a second operation, the cube of this root is
21035·318645155832, and the process by No. 3 of the
Rule will be
21035·318645,&c.
2
42070·63729021035·8
21035·821035·318645, &c.
As63106·43729:diff. ·481355:: 27·6047 :
: the diff. ·000210560
consequently the root required is27·604910560