a aa = 3 + 33 + 2/۳۳۶ في 1 Simple Powers. And therefore it is, that to supply the Want of those Powers (above ce in the Theorem) the Operation mult beropeated : as in the Example of extracting the Cube Root, Page 133, viz. when the Figures in the Root confift of more than three Places (vide Page 140, and 141.) Suppose a a atba=G. Quære a. be G 4+39 5iro+ib tretzitue= 3 G 31 D Which gives this Theorem r+ 31 But if r be taken greater than just, then it will be ret 3r G -60=úrrtib =D, which produces this Tbeoren 31 31 By either of these two Theorems the Value of a may be eafily found. Or rather otherwise, as in the following Example. Let aaa + 24 a=587914. Here b=24. Suppose the first r=90, then y} = 7290007 587914 without the 24 x90 being added to it: Therefore a 590. Again, Suppose r=80 then y? =512000, and 245= 1920. But 512000 +1920=513920 558791, hence 770, but nearer, to it than go." Therefore it must be 11e te= a less than just. I Q3 2 rort3rret 3 ret =ana I x 24 31241 to 241 = 24 2 in Numb. 41512000 -+- 192000 + 24061 = a 3 in Numb. s 1920 + 241 = 246 4 +5 6513920 + 19224 + 240 66 = 587914 D Operation 89 e Operation 80,1 +3 80, is Divisor 83,1) 308,31 3,68 &c. = to= 3,6 249,3 83,68 &c. =rto 2 Divisor 86,7) 59,01 ,67 52,02 87,37) 6,99 &c. Or rather new r=83,7 for a second Operation, which being involved and tried (as above) will be found greater than just: therefore it must be Irla 2r-3rret3reea da 1 x 24 3241 — 246 = 24 a 2 in Numb. 41586376,253 - 21017,07 et 251,111 =aac 3 in Numb. 5 2008,8 241 244 6 + 7121041,072-251,1= 471,053 D 83,7955 - 83,70000000=r i ft Divisor 83,7755) 1,87595778 00,02239331 = 4 ,022 1,675510 83,67760669 = = 2d Divisor 83,7535) ,2004477 ,0023 ,1675070 3d Divifor 83,7512) ,03294078 3 &c. 02512536 83,75? ,00781542 00753760 Here the new Divisors are 27782 rejected, as infignificant. 25125 2657 2512 145 83 8 = 9 All the remaining Examples of extracting Roots (except Page 360) are left in the Author's own Method; which by this Time, it is presumed, the Learner will eafily know how to correst of himklf, if he takes due Notice of what has been delivered Page 131, 132,3". But But if more Exactness be required, you may make the new r= 83,6776067, and proceed with it to a third Operation; which will afford twenty-seven Places of Figures for the Value of a; that is, every Operation will produce triple the Places of Figures to those of the precedent r. And this tripling the Places of Figures in the Root, at every Operation, holds good, and is to be observed in the Solution of all Adfected Equations (how high low ever they are) according to this Method of resolving them. See Page 141. Example 2. Suppose a a a-ba=G. Quære a. If rte {be I a, then qe te 4+6-*r=D, Or you may proceed otherwise, as in the laft Example. Let ааа - 6438 a 104785688, here b = 6438. Suppose the firft r = 500, rrr = 125000000, and.br = 3219000, then 125000000 3219000 = 121781000. But 121781000 7 104785688, therefore r < 500. Again, fuppose r = 400, ror=64000000, and br=2575200, then will 64000000 - 2575200 = 6142800. But 61424800 5104785688, hence 5 7 400; consequently is betwixt 400 and 500. But goo is the next neareft; therefore, let r= 500 being greater than just. Then ifrera 2 rrr-317+3rei= a a a 3 16.rabesba 2 in Numb. 125000000 7500002 to 1500 ee saaa 5 3219000 6438 e = 6438 a (88 4-5 6121781000.- 743562 et 15001 = 1047856 6+71743562 1500 21 = 16995312 7 • 1500 8 495 € -ee=11330 =D D : 495 Operation 3 in Numb. = .او : 8 472) Operation 495 = 20 500,0 =1 1 Divifor 475) 11330 23,8 56 476,2 Śroša 414,0 377,6 Let new r = 476 for a 2d Operation, then 13 = 107850196 and br = 3064488: but 107850176 3064488 = 104785688 the fame with the Refolvend. Consequently a = 476 juft. Example 3. Let ba-041=G. Quære a. Ifrate=, then $ crimin=+rr-}b=D, which gives this D But if ra, then ro Theor am Or otherwise as before in the two laft Examples. Thus, let 123456a-aaa=12272861. Here b=123456. Suppose the firft >= 200, then rrr =8000000, and br=2469 1200; then 24691200-8000000=16691200, but 16691200712272861, therefore r is here less than juft, because the highest Power is or Negative. Again, Suppole r = 300, then y3 = 27000000; and br=37036800, then 37036800 ~ 27000000=10036800 512272861. Consequently 300, and r7 260. Letr = 300, being the next nearest, but more than juft. Then I rea = 3 br-beba 2 in Numb. 4/27000000 - 270000 i-t 90016 3 in Numb. 3 37036800- 123456 5-41 610036800 + 1465448 -- 900 =12272861 6. 1465442-9001 = 2236061 7 + 900 811628–4= 2484 =D D 1* &c. 9 162 Operation Operation 162 = 10 ift Divisor 152) ad Divisor 646) 300,0 = (30 283,4 =r-ero Or new r=283, which being involved, &c. will appear to be the true Root, that is, a = 283 just. Note, These are usually called the three Forms of Cubick Equations; and in the Solution of the third or last Form, viz. ba aaa=G, you may meet with some seeming Difficulties; especially in making Choice of the first r, because this Equation is an ambiguous Equation, and hath ewo Affirmative Roots, viz. a greater and lefser Root. But having once found either of them, the other may be easily obtained by Division only; as in the Quadratick Equations. Vide Chap. 8. As for instance, in tbe laft Example, a=283 and 123456e-aaa=12272861. Make these two Equations = 0, to wit, let a 283= 0, and -aaa+123456 - 12272861 = 0. Then, a 283) - baat 123456 – 12272861 (-a -aaa -+283 a a 2830 2 + 123456 a (-2830 - 283aat 80089 a +43367 a - 12272861 (+43367 + 43367 2-12272861 (0) (0) Hence it appears that al-2832 +4336730. Consequently a a + 283 a=43367 this Equation being solved, a = 110, 2722&c. which is the leffer Root of the aforesaid Equation be -aaa-G, &c. After this Manner all the poffible and impoffible Roots of any Equation may be easily discovered, any one of it's Roots being once found. I shall therefore omit inferting more Examples of that kind. Suppose saatbaa tre=G. Quære a. Let b=74,(=8729, and G=560783. By Trial (as before) it will be found that the next nearest r = 40 being something less than just. Therefore |