only to equations of particular kinds, all of which taken together, form but a small part of the numerous kinds and endless variety of algebraic problems, which may be proposed. But as we have no general rules whereby the roots of high equations can be found, we must be content to approximate as near to the required root as possible, when it cannot be found exactly. 60. The methods of approximation are general, including equations of every kind and description, applying equally to the foregoing equations, and to all others which do not come under the preceding rules: hence approximation is the most general, easy, and useful method of discovering the possible roots of numeral equations, that can be proposed. 61. It must be observed, that one root only is found by these methods, and that not exactly, but nearly. We begin by making trials of several numbers, which we judge the most likely to answer the conditions of the proposed equation; then, (by a process to be described hereafter,) we find a number nearer than that obtained by trial; we repeat the process, and thereby obtain a number nearer than the last; again we repeat the process, and obtain a number still nearer, and so on, to any assignable degree of exactness. 62. The simplest method of approximation. RULE I. Find by trials a number nearly equal to the root of the proposed equation. II. Let r=the number thus found, and let z=the difference between r and the root x of the equation: so that if r be less than x, then r+z=x; but if r be greater than x, then r—z=x. III. Instead of x in the given equation, substitute its equal r+z, or r—z, (according as r is less or greater than x,) and a new equation will arise, including only z and known quantities. IV. Reject every term in this equation which contains any power of z higher than the first, and the value of z will be found by a simple equation. V. If the sign of the value of z be+, this value must be added to the value of r; but if -> it must be subtracted, and the result will be nearly equal to the root required. VI. If this root be not sufficiently near the truth, let the operation be repeated; thus, instead of r in the equation just now resolved, substitute the corrected root, and the second value of z being added or subtracted according to its sign, a nearer approximation to the root will be had; and if a still nearer approximation be required, the operation may be repeated at pleasure, observing always to substitute the last corrected root for the new value of r. EXAMPLES.—1. Given x2+x=14, to find x by approxi mation. By trials it soon appears that x must be nearly equal to 3; let therefore r=3, and r+z=x; wherefore substituting this value of x in the given equation, it becomes r+z+r+z=14, that is, r2+2rz+z2+r+z=14; whence by transposition, and rejecting 14-r-r 14-9-3 2r+1 z2, we obtain 2 rz+z=14—r2—r, and z= 6+1 2 =.28, and x=(r+z=3+.28=) 3.28, nearly. 7 For a nearer value of x, let the operation be repeated. -.0384 7.56 =)—.00508, nearly; wherefore x=(r+z=3.28— .00508) 3.27492, extremely near. 2. Let x3-2x2 +3x=5 be given, to find x. It appears by trials, that x=3 nearly, wherefore let r=3, and r+z=x as before; then will From which, rejecting all the terms which contain z2 or z3, we obtain (r3 +3r3z—2 r2—4rz+3r+3z=5, or) 3r2z−4rz+ a Sometimes it happens that the correction consists of several figures; in that case, if a second operation be necessary, it will be convenient not to substitute all the figures for r, but only one figure, or two, such as will nearly express the value of the whole: thus, if x after the first operation be 3.58, for a second operation I will put r=( =(not 3.58, but) 3.6; if at the conclusion of this second process x= =3.648917, and a third be deemed necessary, I will not employ all these figures, but instead of them put r=3.65, and proceed. This method is to be attended to in all cases, as it saves much trouble, and produces scarcely any effect on the approximation. Let r=2.3, this value substituted for r in the preceding 5-12.167+10.58-6.9 equation, we have z=( 15.87-9.2+3 -3.487 9.67).36, whence x=( (2.3—.36=) 1.94, still nearer than before; and if 1.94 be substituted for r in the equation above alluded to, a third approximation will be had, whereby a nearer value of x will be obtained. 3. Given x2-5x=31, to find x. Ans. x 8.6032778. Ans. x 5.403125. 4. Given x2+2x-40=0, to find x. 5. Given x3+x2+x=90, to find x. 6. Given 2x+4x2-245x-70=0, to find x. Ans. x=10.265. 7. Given x-12x+7=0, to find x. Ans. x 2.0567. 8. Given x2+10x-20=0, to find the value of x. 63. The following method affords a swifter approximation to the unknown quantity than the former rule. RULE I. Let a number be found by trials nearly equal to the required root, and let z=the difference of the assumed number and the true root, as before. This method is given by Mr. Simpson in p. 162. of his Algebra, where he has extended the doctrine beyond what our limits will admit the above rule is in its simplest form, and triples the number of figures true in the root, at P every operation; he calls it an approximation of the second degree, (z= a α if the first value of z (viz, be substituted in the second term of the denominator, and the following terms be rejected, it will become z= ap an approximation of the second de a2 + bp' bq2 gree, the same as the above rule. If for z its second value 9 be substi α an approximation of the third degree, which II. Substitute the assumed quantity +z, in the given equation, as directed in the preceding rule; and the given equation will be reduced to this form, az+bz3+cz3 +, &c.=p. III. By transposition and division we have z= &c. where, if all the terms after the first be rejected, we shall have z ; and if q be put for 2, α and its square substituted α for z2 in the second term, we shall have z= EXAMPLES.-1. Given x3-2 x2+3x=5, to find x. Here x=3 nearly; let 3+z=x, then, x3= 27+27 z+9 z2+z3 -2x=-18-12 z-2 z2 =5, that is, 18+18z+722+z3=5, or 18 z+7 z2+z3 =—13. Here a=18, b=7, c=1, p=—13, q=(-2—=—-13=)—.72, a 18 and rejecting bs2q3 (as very small) from the product, becomes a+sp.p a2 + b + as.p the approximating rule of the fourth degree is ap.a+ wp ber of figures true at every operation. which quintuples the num Here a=7, b=4, c=1, p=−1, and q=(? q=(2=====)— α And x=(2.0784-.15451064) 1.92388936, very nearly. 2. Given x2+20x=100, to find the value of x. 4.1421356. 3. Given x3-2x=5, to find r. Ans. x 2.094551. Ans. x= 4. Given x-48 x2+ 200=0, to find x. Ans. x= 47.91287847478. 5. Given x4-38 x3 +210 x2 +538x+289=0, to find x. Answer, x=30.5356537528527. 6. Given x+6 xa — 10 x3 — 112 x2-207 x-110=0, to find x. Ans. x=4.4641016151. 7. Given 2x2+3x+4=50, to find the value of x. 64. BERNOULLI'S RULE Has been sometimes preferred on account of its great simpli city and general application: it is as follows. RULE I. Find by trials, two numbers as near the true root as possible c. This is perhaps the most easy and general method of resolving equations of every kind, that has ever yet been proposed; it was invented by John Bernoulli, and published in the Leipsic Acts, 1697. The most intricate and difficult forms of equations, however embarrassed and entangled with radical, compound, and mixed quantities, readily submit to this rule without any previous reduction or preparation whatever; and it may be conveniently employed for finding the roots of exponential equations. The rule is founded on this supposition, that the first error is to the second, as the difference between the true and first assumed number is to the difference between the true and second assumed number: and that it is true according to this supposition, may be thus demonstrated. Let a and b be the two suppositions; A and B their results produced by similar operations; it is required to find the number from which N is produced by a like operation: in order to which, Let N-A=r, N—B=s, and x = the number required; then by hypothesis, r: s:: x-a; x-b, whence dividendo r-s: s:: b—a: x-b, that is, b-a.s =x-b, which is the rule when both the assumed quantities, a and b, are less than the true root x. |