But if the given Quantity hath not fuch 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 xx, very near. Now if by this Divifor, you Divide the next foregoing Refol + ༡༢༠ 21212 + 33214 2048, 's ༢་ for the Value of the next 7, which added tor + _____ 27 - 8r3 +, the Value of the next foregoing, is near the 16rs' Square-Root of r2 ± z2. N. B. The Sign + denotes or and the Sign denotes or t. Corollary. Here you may fee that the Uncie of the Square-Root of a Binomial or Residual, are —1, 1, 1× 1 = I X 2 =2&; that is, putting n Index of the Square Root or, the Uncie of the Square Roor of a Binomial or Residual 3. Let it be required to Extract the Cube-Root of r* ± 23, very near. *+ ›××+7×*+y=xxx+ 3xxx,+ 3×9 †ÿ Canon. Here you may fee that the Uncia of the Cube-Root of a Bino mial or Refidual are 1, 3, 3× In infinitum; that is, putting n the Uncie of the Cube Root of a n 2 I -2 &c. 2 3 Index of the Cube-Root) Binomial or Refidual, are = 1, n- I nx 3 In like manner you may find that putting n. the Uncia of the Biquadrat-Root of a Binomial or Refidual, are =1,n,nX tum. And fo on for fuperiour Roots; whence, and from what has been faid in Part 11. Chap. I. we have good Reason to believe that. Universally the Uncia of any Binomial or Residual, whofe Index is≈n, are 1, n, nx X ×2 — 2×2———3, &c. But for a further Confir n n n- 3 mation of this, I refer you to Part XV. Chap. I. CHA P. II. and III. Evolution and mixt Evolution of Pumbers. NO Ote, What I call mixt Evolution, is the Method of Extra- Hule. When you are to Extract any urmixt Root; viz. the SquareRoot, Cube-Root; &c. of a given Number; Involve any Bino mial into it felf 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 Extract any of the Roots of an adfected Equation, fuppofe the Binomial x+y= the Root you feek; then inftead of the faid Root (or unknown Letter) and its Powers in the Æquation, fubftitute their respective Values, vixx+y and the respective Powers thereof. And the Sum of the Terms wherein either x or y occurs, in the Equation thus had, is your Canon for Evolving the abfolute Number in this (or in the propos'd) Equation. Having thus fram'd a Canon for Evolving, the Operation is to be perform'd in the following Manner, viz. ift. Find the firft or greateft Member (viz. the first Significant Figure, with its due number of Cyphers) of the Root fought; and call it x; then having found the Value of the firft Term, or of the Sum of the firft Terms of your Canon, i. e. of all those Terms wherein x and its Powers only occur, Substract it from the absolute Number; the Remainder call your Refolvend. And the Value of the Coefficient, or of the Sum of the Coefficients of y in the second Term or Terms of your Canon, call your Divifor. Now by Dividing the faid Refolvend by this Divifor, the Value of 3, or the 2d. Member of the faid Root is in fome Cafes found,but not in all; wherefore in the beginning of your Operation, you muft 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 not exceed the faid Refolvend. Having thus found the Value of y, as alfo the Sum of the Values of all the Terms of your Canon, wherein y occurs; pláce the former along with the before found Value of x in the Roor, and call the latter your Ablativum ; which Substract 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 x, (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 Refolvend leaves o; or 'till you have as many Decimal Figures as you think fufficient, Note, tho' by the first Divifion, you may not find the next Member of the Root fought; yet in continuing the Operation, one Division may ferve to find feveral of the next following Members, or the Value of 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. |
