'r §8 A 'Treatise of Algebra.

cording to the first Theorem, and errs only in the lastPlace: But according to the second Theorem, the An-

554 +

swer will come out 14 -j-= 1.4.14.212*62x6*

39005 3

which is true to the last Place: And, if with this lastNumber the Operation, be repeated you will have theAnswer true to nearly 60 Places of Decimals.

Example 2. Let it be required to extract the CubeRoot of 1728. Let r be taken = 11, then k being

— 1728 and » —z, we shall have v (———A —

\i — r"/

3993

— = 10.05793, and therefore r -f*

397 _

rx»+s .

— ' -.. . r — - — - — 1 *>9999* i

2v-j- zn — 1 x 1/ ~ x » — i x 2 n — 1

which differs from Truth by only -Part of an,

50000

Unit.

Example 3. In which it is proposed to extract theCube Root of 500. Here the required Root appearingto be less than 8, but nearer to 8 than 7, let r be taken,

— 8, and we shall have v (— ^^— ,28; and

— 1 2 '

therefore r ■

rX2v-\-n

2v-\- zn — 1 Xv^-~rX n — 1 x 2 n —1

6072

— 8---— 7.937005259936 3 which is true

96389

to the last Place.

Example 4. Lastly, let it be proposed to extract thefirst furfolid Root of 125000. In this Cafe we havek =125000, n=5, r=io, and v =: 20 ; whencethe required Root ‘yvill be found 40.456389.