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 BRA. 87 A L G 2 E the root; fubtradl its fquare from that part, and anThe fqiure 1 root of any compound quantity, as « + nex the fecond part of the given number to the re2 a b + b is difcovered after this manner. “ Firil:, mainder. Then divide this new number (negledting “ take care to difpofe the terms according to the dimen- its laft figure) the double of the firft figure of the “ (ions of the alphabet, as in divifion ; then find the root, annex thebyquotient double, and multiply “ fquare root of the firft term a which gives a for the the number thence arifingtobythat the faid quotient, and if “ firfi: member of the root. Then Ihbtradt its fquare the product is lefs than your dividend, to it, “ from the propofed quantity, and divide the firfl: term that quotient lhall be the fecond figureorofequal the root. « of the remainder by the double of that mem- But if the product is greater than the dividend, you “ ber, viz., 2a, and the quotient Ms the fecond member muft take a lefs number for the fecond figure of “ of the root. Add this fecond member to the double the root than that quotient.” Much after the fame “ of the firft, and multiply their fum {za+b), by the femay the other figures of the quotient be found, “ cond member b, and fubtraft the product (2<jA+^2) manner there are more points than two placed over the given “ from the forefaid remainder (2«i+^1) and ifmothing ifnumber. “ remains, then the fquare root is obtained;” and in this To find the fquare root of 99856, firft point it thus, example it is fouud to be « + A The manner of the operation is thus, 99856; then find the fquare root of 9 to be-3, which a2+2ab+bz {a+b therefore is, the firft figure of the' root; fubtradf 9, the fquare of 3, from 9, and to'the remainder annex the fecond part 98, and divide (negledling the laft figure 8) by 2a--bab'-b'21 the double of 3, or 6, and place the quotient after 6, y.b) 2ab--b . and then multiply 61 by 1, and fubtradl the produdt 61 from 98. Then to the remainder (37) annex the laft 0.0 part of the propofed number (56), and dividing 3756 (nethe laft figure 6) by the double of 31, that is by “ But if there had been a remainder, you muft have gleding 62, place the quotient after, and multiplying 626 by the “ divided it by the double of the fum of the two parts quotient you will find the produdt to be 3756, which “ already found, and the quotient would have given the fubtradted6,from the dividend, and leaving no remainder, “ third member of the root.” x the exadt root muft be .316. Thus, if the quantity propofed had been a --2abAr 2&c--b2--2bc-'c2, after proceeding as 2above, you would Examp. 99856 (316 have found the remainder 2ac--2bc--c, which divided _.9_ by 2a--2b gives e to be annexed to a--b as the gd mem61x98 ber of tb£ root. ThenJ adding eto 2a--2b, and multiplyX1J61 ing their2 fum 2a--2b rc by c, fubtradt the produdt 2^e-(626x3756 zbc+c from the forefaid remainder ; and fince nothing x6/3.7:56_ now remains,, you conclude that a+b+c is the fquare root required. o The fquare root of any mTmber is found out after the to extradt any root out of any given quanfame manner. If it is a number under ioo, its neareft tity,In general, “ Firft range that quantity to thedimenfquare root is found by the-following table ; by which al- “ fions of its letters, and extradt according the faid root out of the fo its cube root is found if.it be under iooo, and is bi- “ firft term, and that fhall be the firft member of the quadrate if it be under ioooo. “ root required. Then raife this root to a dimenfion 8 “ lower by unit than the number that denominates the jThe rootj required, and multiply the power that arifes by [Square J _LI_4 _9! Jbj Ji' 361 ■ 4.9]'?l 64 81? “ root that number itfelf; divide the fecond term of thq jCube i llJ: 27| 64I125I 216 343| 51-2 729 ““ given by the produdl, and the quotient fliall (Biquad.j 1 116 8ij256|6g5i 12961 24011 4096 6?6i “ give thequantity fecond member of the root required.” y to3extradl the root of4 thes 5th power out of/7 -|1 But if it is a number above loo,, then its fquare root ^a4Thus b-i-ioa b +ioa2b3-j-sab +b, I find, that the root will confift of two or more figures, which mull be found of the 5th power out of as gives a, which I raife to the by different operations by the following 4th power, and multiplying by 5, the produdt is 5a4 ; then dividing the fecond term of the given quantity R U L E. 4 I find b to be the fecond member ; and raiftng, “ To find the fquare root of any number, place a by 5«tov the 5th power, and fubtradting it, there being no “ point above the number that is in the place of units, a+b remainder, I conclude that a-j-b is the root required. If “ pafs the place of tens, and place again a point over that the has three members, the third is found after the, “ of hundreds, and goon towards the left hand placing a root manner from the firft two confidered as one mem“ point over every 2d figure; and by thefe points the fame as the fecond member was found from the firft “ number will be dillinguifhed into as many parts as ber, may be eafily underftood from what w as faid of “ there are figures in the root. Then find the fquare which “ root of the firll part, and it will give the firft figure of extradling the fquare root. In