Substituting 3 for x, in the above functions, we have

+0 variation.

x=4 + + +

Hence there are two roots between 3 and 4.

As the sum of the roots must be -11, and the two positive roots are more than 6, there must be a root near -17.

As there are two roots between 3 and 4, we will transform the equation, (Art. 175.), into another, whose roots shall be 3 less; or put x=3+y. Then we shall have

The value of y, in this transformed equation, must be near the value of y in the equation 122y=27, (Art. 183.); that is, y is between .2 and .3

y=.2 gives +

+ 2 variations.

y=.3 gives + + + + O variation.

Hence there are two values of y between .2 and .3; and, of course, two values of x between 3.2 and 3.3.

We may now transform this last equation into another whose roots shall be .2 less, and further approximate to the true values of x, in the original equation.

Having thus explained the foregoing principles, and, in our view, been sufficiently elaborate in theory, we shall now apply it to the solution of equations, commencing with

NEWTON'S METHOD OF APPROXIMATION.

(Art. 189.) We have seen, in (Art. 175.), that if we have any equation involving x, and put x=a+y, and with this value transform the equation into another involving y, the equation will be

If a is the real value of x then y=0, and X=0.

If a is a very near value to x, and consequently y very small, the terms containing y2, y, and all the higher powers of y, become very small, and may be neglected in finding the approximate value of y.

In the equation x=a+y, if a is less than x, y must be positive; and if y is positive in the last equation, X and X' must have opposite signs, corresponding to (Art. 184.).

Following formula (1), we have an approximate value of y; and, of course, of x. The value of x, thus corrected, again call a, and find a correction as before; and thus approximate to any required degree of exactness.

EXAMPLES.

1. Given 3x+4x3-5x-140-0, to find one of the approximate values of x.

By trial we find that x must be a little more than 2.

X= 3(2.07)5+ 4(2.07)3—5(2.07)—140...By log. X =—0.854

8.54

Hence the second value, or Y=3218.2 0.00265+

2. Given x+2x2-23x=70, to find an approximate value

of x.

Ans. 5.1345-+.

3. Given 2-3x2+75x=10000, to find an approximate value of x. Ans. 9.886+.

4. Given 324-35x3-11x2-14x+30=0, to find an approximate value of x. Ans. 11.998+.

5. Given 5x3-3x2-2x=1560, to find an approximate value 7.00867+.

of x.

Ans.

CHAPTER VI.

HORNER'S METHOD OF APPROXIMATION.

(Art. 190.) In the year 1819, Mr. W. G. Horner, of Bath, England, published to the world the most elegant and concise method of approximating to roots of any yet known.

The parallel between Newton's and Horner's method, is this; both methods commence by finding, by trial, a near value to a root.

In using Horner's method, care must be taken that the number, found by trial, be less than the real root. Following Newton's method we need not be particular in this respect.

In both methods we transform the original equation involving x, into another involving y, by putting a=r+y, as in (Art. 175), r being a rough approximate value of x, found by trial.

The transformed equation enables us to find an approximate value of y, (Art. 189.).

Newton's method puts this approximate value of y to r, and uses their algebraic sum as r was used in the first place; again and again transforming the same equation, after each successive correction of r.

Horner's method transforms the transformed equation into another whose roots are less by the approximate value of and y; again transforms that equation into another whose roots are less, and so on, as far as desired.

By continuing similar notation through the several transformations we may have

x=r+y

y=8+z

z=1+z'

`z'=u+z"

&c. &c.

Hence xr+s+t, &c.; r, s, t, &c., being successive figures of the root. Thus if a root be 325, r=300, s=20, and t=5. On the principle of successive transformations is founded the following

RULE for approximating to the true value of a real root of an equation.

1st. Find by Sturm's Theorem, or otherwise, the value of the first one or two figures of the root, which designate by r. 28. Transform the equation (Art. 175), into another whose roots shall be less by r.

3d. With the absolute term of this transformed equation for a dividend, and the coefficient of y for a divisor, find the next figure of the root.

4th. Transform the last equation into another, whose roots shall be less, by the value of the last figure determined; and so proceed until the whole root is determined, or sufficiently approximated to, if incommensurable.

NOTE 1. In any transformed equation, X is a general symbol to represent the absolute term, and X' represents the coefficient of the first power of the unknown quantity. If X and X' become of the same sign, the last root figure is not the true one, and must be diminished.

NOTE 2. To find negative roots, change the sign of every alternate term, (Art. 178.): find the positive roots of that equation, and change their signs.

(Art. 191.) We shall apply this principle, at first, to the solu tion of equations of the second degree; and for such equations as have large coefficients and incommensurable roots, it will furnish by far the best practical rule..

Y

EXAMPLES.

1. Find an approximate root of the equation

x2+x-60=0.

We readily perceive that x must be more than 7, and less than 8, therefore r=7.

Now transform this equation into another whose roots shall be less by 7.

Operate as in (Art. 175.), synthetic division

Here we find that y cannot be far from 5, or between .2 and .3; therefore transform the last equation into another whose roots shall be .2 less; thus,

.96

To obtain an approximate value of z, we have

In being thus formal, we spread the work over too large a space, and must inevitably become tedious. To avoid these difficulties, we must make a few practical modifications.

1st. We will consider the absolute term as constituting the second member of the equation; and, in place of taking the algebraic sum of it, and the number placed under it, we will take their difference.