purpofe; one of the moft fimple is that of Sir Ifaac Newton, which fhall now be explained.

LEMMA. If any two numbers being inferted for the unknown quantity (x) in any. equation, give refults with oppofite figns, an odd number of roots must be between these numbers.

This appears from the property of the abfolute term, and from this obvious maxim, that, if a number of quantities be multiplied together, and if the figns of an odd number of them be changed, the sign of the product is changed. For, when a positive quantity is inferted for x, the refult is the abfolute 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 refult has the fame fign as the given abfolute term, then from the property of the abfolute term, (Prop. 2. Chap. 1.) either none, or an even number only, of the positive roots have had their figns changed by the transformation; but, if the refult has an oppofite fign to that of

the

the given abfolute term, the figns 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 fitive roots of the given equation, or less than any of them; in the fecond cafe, it must have been greater than each of an odd number of the pofitive roots. An odd number of the pofitive roots therefore must lie between them, when they give results with oppofite figns. The fame obfervation is to be extended to the fubftitution 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 firft ftep 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 x-2x-5=0.

1. Find the nearest integer to the root; in this cafe a root is between 2 and

3,

for

thefe

thefe numbers being inferted for x, the one gives a pofitive, and the other a negative refult. Either the number above the root, or that below it, may be affumed as the first value; only, it will be more convenient to take that which appears to be nearest to the root, as will be manifeft from the nature of the operation.

2. Suppofe x=2+f, and fubftitute this value of x in the equation.

3

x3—2x−ƒ=—1+10ƒ+6ƒ" +ƒ3 =0.

If ƒ is less than unit, its powers f2 and f3 may be neglected in this first approximation, and 10f=1, or f=0.1 nearly, therefore x = 2.1 nearly.

3. As f=0.1, nearly, let f=.1+g, and infert this value of ƒ in the preceding equa

ƒ3+6ƒ3+10ƒ—1= 0.061+11.238+6.38*+g3=0

and

and neglecting g2 and gas very small 0.61

+11.2380, or g=

11.23

-.0054,

hence f=0.1+8=.0946 nearly, and x= 2.0946 nearly.

4. This operation may be continued to any length, as by supposing g=-.0054+b2 and so on, and the value of x=2.09455147 nearly.

By the first operation a nearer value of y may be found thus; fince f. I nearly, and ·1+10ƒ+6ƒ2 +ƒ =0,

I

I

ƒ=10+0+ that is ƒ=10+.6+.01

=.094

true to the laft figure, and x=2.094. In the fame manner may the root of a pure equation be found, and this gives an eafy 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 affuming a general equation, general rules may be deduced for approximating to the roots of any proposed equation. By a fimilar method we may approximate to the roots of

literal equations, which will be expreffed by infinite feries.