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purpose; one of the most simple is that of Sir Isaac Newton, which shall now be explained.

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LEMMA. If any two numbers being inserted for the unknown quantity (x) in any equation, give results with opposite signs, an odd number of roots must be between these numbers.

This appears from the property of the absolute term, and from this obvious maxim, that, if a number of quantities be multiplied together, and if the signs of an odd number of them be changed, the sign of the product is changed. For, when a positive quantity is inserted for x, the result is the absolute term of an equation whose roots are less than the roots of the given equation by that quantity, (Prop. 2. Cor. 3. Chap. 2.). If the result has the same fign as the given absolute term, then from the property of the absolute term, (Prop. 2. Chap. 1.) either none, or an even number only, of the positive roots have had their signs changed by the transformation; but, if the result has an opposite' sign to that of

the

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the given absolute term, the signs of an odd number of the positive roots must have been changed. In the first case, then, the quantity substituted must have been either

greater than each of an even number of the po. sitive roots of the given equation, or less than any of them; in the second cale, it must have been greater than each of an odd number of thé positive roots. An odd number of the positive roots therefore must lie between them, when they give results with opposite signs. The same observation is to be extended to the substitution of negative quantities and the negative roots.

From this Lemma, by means of trials, it will not be difficult to find the nearest integer to a root of a given numeral equation, This is the first step towards the approximation; and both the manner of continuing it, and the reason of the operation, will be evident from the following example.

Let the equation be *--2x–5=0.

1. Find the nearest integer to the root; in this case a root is between 2 and 3, for

these

x

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these numbers being inserted for x, the one
gives a positive, and the other a negative
result. Either the number above the root,
or that below it, may be assumed as the first
value; only, it will be more convenient to
take that which appears to be nearest to the
root, as will be manifest from the nature of
the operation.

2. Suppose x=2 tf, and substitute this
value of x in the equation.

x?= 8+12f+6ff+f?
-2x =-4-2f

5 =-5
x?—2x—f=-it1of+6f2+fco.

If f is less than unit, its powers f? and
f' may be neglected in this first approxi-
mation, and 10f=1, or fro.I nearly,
therefore x=2.1 nearly.

3. As f=0.1, nearly, let f=.i+g, and
insert this value of f in the preceding equa-
tion.

f3= 0.001 +0.038+0.3g2+g3
Of ?= 0.05 + 1.2g+ 6.g?
10f=

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I

+ 10g

I=MI

f}+6f* +10f-1= 0.061+11.238+6.38*+go=0

and

-0.261

11.23

.

I

fri

and neglecting 32 and go as very small 0.61 + 11.238=0, or g= =-.0054, hence f=0.1+g=.0946 nearly, and x= 2.0946 nearly.

4. This operation may be continued to any length, as by supposing g=-.00;4+b, and so

on,

and the value of x =2.09455147 nearly:

By the first operation a nearer value of x; may be found thus; since f=.I nearly, and -1 +10f+6f2+f=o,

that is f=107.67.01=.094 10+68+82) true to the last figure, and x=2.094.

In the same manner may the root of a pure equation be found, and this gives an easy method of approximating to the roots of numbers, which are not perfect powers.

This rule is applicable to numeral equations of every order, and, by assuming a general equation, general rules may be deduced for approximating to the roots of any proposed equation. By a similar method we may approximate to the roots of literal equations, which will be expressed by infinite series.

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