Now, if by this Divifor, you Divide the next foregoing Re folvend, you'll find 528 7210 + 33214 128r7 25619 1024r11 2048r13 for the Value of the next y, which added to r± 26 27 2.4 823 + the Value of the next foregoing x, is nearly the 16rs' Square-Root of rr ± zz. Corollary. Here you may fee that the Uncia of the Square-Root of I a Binomial or Refidual are equal to 1,, × X ; &c. that is, putting n = Index of the Square 2 3 Root, or, the Uncie of the Square-Root of a Binomial or 3. Let it be required to Extract the Cube-Root of r3 ±z3 nearly. :x+y:x:x+y:x:x+y=x3+ zxxy + zxyy +y3. Ca non, Operation. 529 243r 22%1s 729r14 In this laft Divifor x is 229 729r's Refolvend. Z12 = 3xx 9r5 by which Divisor, if you Divide its respective Refolvend, you'll have ± 8178 therefore the Cube-Root of r3+z3 is nearly r+ H 81752 + 9r1 + 3rr for the Value of the next y. And 23 3rr Here you may fee that the Uncia of the Cube-Root of a Binomial or Refidual are 1, +× I 1483 2 X 2 3 &c. In infinitum; that is, putting n Index of the CubeRoot, the Uncie of the Cube-Root of a Binomial or Re In like Manner you may find that putting n=4, the Uncie of the Biquadrat-Root of a Binomial or Residual are equal to ท. I n - I, n, nx 2 ท- -3 X 2 2 3 2 &c. In infinitum: And fo on for fuperiour Roots. From what has been faid in Chap. 1. Part 3. and in the precedent Corollaries and Scholium, we have good Reafon to believe that Universally the Uncie of any Binomial or Residual, whose Index is n,are equal to 1,n,n x N-- I n2 n 3, c: But for a further Confirma tion of this, I referr you to Part XV. Chap. 1. N CHA P. II, III. Evolution, and mixt Evolution of Numbers. Ote, What I call mixt Evolution, is the Method of Extracting the Roots of adfected Equations. Kule. When you are to Extract any unmixt Root; viz. the SquareRoot, Cube-Root, &c. of a given Number; Involve any Bino mial, as xy 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 instead of the faid Root (or unknown Letter) and its Powers in the Equation, fubftitute their respective Values, viz. x+y, and the refpective 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. ift. Find the firft or greatest Member (viz. the firft fignificant Figure, with its due Number of Cyphers) of the Root fought; and call it x; then having found the Value of the first Term, or of the Sum of the firit Terms of your Canon, i. e. of all those Terms wherein x and its Powers only occur, Subtract it from the abfolute Number; the Remainder call your Refolvend: And the Value of the Co-efficient, or of the Sum of the Co-efficients of in the fecond Term or Terms of your Canon, call your Divifor. Now by Dividing the Refolvend by this Divifor the Value of y, or the fecond Member of the Root is found in fome Cafes, 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 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; place the former along with the before found Value of x in the Root, and call the latter your Ablativum, which Subtract from the faid Refolvend; and, if there be a Remainder, call it your next Refolvend. The Sum of the Values of the foregoing x and y call x, (i.e. 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 Divifion may serve to find several of the next following Members, or the Value of y to many Places of Figures, as will appear in the latter end of this Part IV. Examples. 1. Suppofe it was required to Extract the Square-Root of 4624. 1st.: x+y:x:x+y=xx+2xyyy is your Canon for Extracting the Square-Root. * in Page 42. 2dly. The greatest Member of the Square- * See the Table Root of 4624 is 60 (for 70 × 70 = 4900) ; therefore x = 60, and xx 3600, which taken from 4624, leaves 1024 for a Refolvend; and the Co-efficient of in the fecond Term of your Canon is 2x=120 for your Divifor; by which Dividing the Refolvend, viz. 1024, the Quotient is 8=y; therefore 2xyyy 1024 is the Ablativum, which taken from the Refolvend, leaves o. Whence the Square-Root of 4624 is 60 +8=68. Operation. = 2. Let it be required to Extract the Cube-Root of 99252847. :*+y:x:x+ÿ:x:x+y:=xxx + 3xxy + 3xyy + 999. Canon. Operation. 99252847 (400+60 +3=463 the Root required. |