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6an 2xy

Herey =

3n.
tonn = уу
ban + gon . Ablativum,
+ 2ax

6nx + xx Resolvend.
Divifor 24 69 2x

SHere x (i, e, 2d. x) 6nx = 2xy

3n. Andy

(i.e. 2d. y) = x . t ** = yy 6nx t ** Ablativum,

Remainder.

20x

2ax

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But if the given Quantity hath not such a Root as is required, then the Evolution may be continued to an endless Series. Thus

2. Let it be required to Extract the Square Root of rr * *x, very near.

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337"*

27

a

Now if by this Divisor, you Divide the next foregoing Rerolvend, you'll find

578
+ 77! 218

+
128 76
256r" 1024,".

20487"} for the Value of the next y, which added tort: 3

34

'8r3 + , the Value of the next foregoing *, is near = thie

1675 Square-Root of r $74

N. B. Tbe Sign + denotes for and the Sign F denotes or to

Corollary. Here you may fee char the Encic of the Square-Root of a Binomial or Residual, are=1,1, x , ; inix = x

를 *2,6; that isy putting n = Index of the Square-Roor

&c or = }, the Unciæ of the Square Roor of a Binomial or Residual are , n, X nxx x *****=3,8c. In infinitum. 2

. 3. Let it be required to Extract the Cube-Root of r: + 7", vety near, ** 3** +,x*+, =*xx+ 3*43+ 389 +m Canon,

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Operation

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visor, if you Divide its respective Resolvend, you'll have 579

1OZ'2 227's +

ot for the Value of the nexty. 81,8 2437!!

,

729r's

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And therefore the Cube Root of 73° + is near = 16 76

107! 227's

+ 3r grs 8178

2437"

7297"}

579

Corollary. : Here you may see that the Uncie of the Cube-Root of a Bino. mial or Refidual are 1, }, { x

キーンさん
Х

*,&c.

3 In infinitum; that is, putting n= Index of the Cube-Root = $. the Unciæ of the Cube Root of a Binomial or Refidual, are =1,

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In like manner you may find that putting n= the Cnciæ of the Biquadrat-Root of a Binomial or Refidual, are =1,9,9 X --

x 239 -3,6c. In infini

3,8c 3 tum. ' And so on for superiour Roots; whence, and from what has been said in Part bil. Chap. 1. we have good Reason to believe that Univerlally the Vnciæ of any Binomial or Residual, whose Index is n, are = 1, n, nx:

3 X 1-2x, 8c. But for a further Confir

&.

4 mation of this, I refer you to Part XV. Chap. I.

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CHA P. II. and Il.
Evolution and mixt Evolution of Rumbers.
Ote, What I call mixt Evolution, is the Method of Extra-

iting the Roots of adfected Aquarions.

Note,

Tule. When you are to Extract any unmixt Root , vit. the SquareRoot, Cube-Root; &c. of a given Number ; Involve any Bino.

mial

H%

mial * ty into it self according to the Number Denominating the Root to be Extracted ; and the Power thus produc'd is your Canon for Evolving. But when you are to Extrad any of the Roots of an adfected Æquation, suppose the Binomial x'ty= the Root you seek; then instead of the said Root (or unknown Letter) and its Powers in the Æquation, substitute their respective Values, viz_xty and the respective Powers thereof. And the Sum of the Terms wherein either x or y occurs, in the Æquation thus had, is your Canon for Evolving the absolute Number in this (or in the propos'd) Æquation.

Having thus fram'd a Canon for Evolving, the Operation is to be perform'd in the following Manner, viz.

ift. Find the first or greateft Member (viz. the first Significant Figure, with its due number of Cyphers) of the Root sought ; and call it x; then having found the Value of the firkt Term, or of the Sum of the first Terms of your Canon, i.e. of all those Terms wherein x and its Powers only occur, Subftract it from the absolute Number ; the Remainder call your Resolvend. And the Value of the Coefficient, or of the Sum of the Coefficients of in the second Term or Terms of your Canon, call your Divifor, Now by Dividing the said Refolvend by this Divisor, the Value of y, or the ad. Member of the said Roor is in some cases found, but not in all ; wherefore in the beginning of your Operation, you must take care that y be the greatest Member, and that the Sum of the Values of all the Terms of your Canon wherein y occurs, may nor exceed the said Resolvend. Having thus found the Value of y, as also the Sum of the Values of all the Terms of your Canon, wherein y occurs; place the former along with the before found Value of x in the Roor, and call the latter your Ablativum ; which Subftract from the said Refolvend; and, if there be a Remainder, call it your next Refolvend. The Sum of the Values of the next foregoing x and y call %, (i. c. a 2d. x nearer the truth). And proceed with this Value of x, in order to find that of the next y as before is Taught. And thus proceed 'till the ablativum taken from its Resolvend leaves o; or 'till you have as many Decimal Figures as you think sufficient.

Note, tho' by tbe first Division, you may not find the next Member of the Root fought ; get in continuing the Operation, one Division may serve to find several of the next following Members, or the Value of ý to many places of Figures, as will appear in the last part of this Part IV.

1. Suppose it was required to Extract the Square Root of 1024.

i

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