dent that the root x (r x 1 + z) of the given equation "k, will be equal to r+"px I + n + 1. ¿?. 6 nearly; But both these theorems will be rendered a little more commodious, by putting v ing~~ nrn and fubftitut - in the place of its equal, p, whence, after proper réduction, x will be had r+ nearly; and equal to r+ more nearly. rx tv + n + 1 v × 6v +4n−2 rx 2v + n v × 2v+2n−1+3.n−1.2/−1 I fhall now put down an example, or two, to fhew the use and great exactness of these laft expreffions. 1. Let the equation given be 22, or, which is the fame, let the fquare root of 2 be required. Then, affuming r 1.4, we have n 2, k = 2, 2 × 1,96 = k-ri 2 - 1.96 98; and therefore r ηγη ( the value of x according to the former approximation; but, but, according to the latter, the answer will come out 1.4 + 5544 39005 1.41421356236; which is true to the laft figure: and, if with this number the operation be repeated, you will have the answer true to nearly 60 places of decimals. 2. Let it be required to extract the cube root of 1728. Here, taking r 11, we fhall have ( nr" ==11,99998; 50000 2v + 2n − 1 × V + 3 × n − 1 x 2n - I which differs from truth by only unit. 3. Let it be propofed to extract the cube root of 500. Here, the required root appearing to be less than 8, but nearer to 8 than 7, let r be taken = 8, and 3 × 512 ) = — 128; and there we shall have v ( = fore r + 12 r x 2v + n 2v + 2n — I × v + / × n — I × 2n - I 6072 =8. 4. Laftly, let it be propofed to extract the first furfolid root of 125000. In which case k being 125000, n = 5, r = 10, and v 20, the required root will be found 10,456389. Befides the different approximations hitherto delivered, there are various other ways whereby the roots of equations may be approached; but, of thefe, none more general, and eafy in practice, than the following. Let and Let the general equation, az + bz2 + cz3 + dxz+ + ez3 &c. = p, be here refumed; which, by divifion, after the first, we shall have x= mation of the first degree. ; and neglect all the terms And if this value of z be now fubftituted in the fecond term, and all the following ones be rejected, we of the fecond degree. ); which is an approximation In order now to get an approximation of the third degree, let this laft value be fubftituted in the fecond term, neglecting all the terms after the third; fo fhall + X + 2 : but here, in the room of 22, either of the fquares of the two preceding values of z, or their rectangle may be fubitituted, that is, either I A A A ; but the last of these A' B B A B =() is the most commodious; whence we have z = Again, for an approximation of the fourth degree, we b lues being fubftituted in the general equation and all the terms after the four first rejected, there now comes out C aС + bB + cA + d In like manner, for an approximation of the fifth de b b C B gree, we fhall have Ż= X X P Ꭰ C LD + 2C + B + @D+ P E = aD+bC+cB÷dA+e. Whence the law of conda te manifeft; tinuation is manifeft; whereby it appears, that if there be D=aC+bB+cA+d, E = aD+bC+cB+ dA +e &c. then will A B C D E F A'B'C' D'E' F'G' many, fucceffive, approximations to the value of z, afcending gradually, from the lowest to the fuperior orders. An example will help to explain the use of what is above delivered; wherein we will fuppofe the equation given to be 12% + 6x2 + x3 = 2. Here a=12, b=6, c=1, d=0, e=0, &c. and p=2; a A + b 12x6+6 )= whence A (==) = 6, B (= aC+bB+ · 505; D ( = aC + b + c A + d) – 6×505+6×39+6 2 Ꭰ 654 From the fame equations the general values of B, C, D, &c. may be eafily found, in known terms, indepen dent of each other. Thus B (: 2ab + + + pxa++3a2bp+2ac+bb . p2 +dp3 a3+4a3bp+3ac + 3bb. ap2+be+ad. 2p3+ep*, &c. which are fo many different approximations to the value of z. Thus far regard has been had to equations which confift of the fimple powers of one unknown quantity, and are no ways affected, either by furds or fractions. If either of these kinds of quantities be concerned in an equation, the ufual way is to exterminate them by multiplication, or involution (as has been taught in Sect. |