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15. To find four numbers in geometrical progression, whose sum is 15, and the sum of their squares 85.

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A CUBIC EQUATION, or equation of the third degree or power, is one, that contains the third power of the unknown quantity as x3-ax2+bx=c.

A biquadratic, or double quadratic, is an equation, that contains the fourth power of the unknown quantity: as 4 x-ax3+bx2-cx=d.

An equation of the fifth power, or degree, is one, that contains the fifth power of the unknown quantity: as as -ax^+ bx3-cx+dx=e.

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An equation of the sixth power, or degree, is one, that contains the sixth power of the unknown quantity as a —ax3-bx1 —cx3 +dx3 —ex=f.

And so on, for all other higher powers. Where it is to be noted, however, that all the powers, or terms, in the equation are supposed to be freed from surds, or fractional exponents.

There are various particular rules for the resolution of cubic and higher equations; but they may be all easily resolved by the following rule of Double Position.

RULE.

RULE.*

1. Find by trial two numbers, as near the true root as possible, and substitute them separately in the given equation, instead of the unknown quantity; marking the errors, which arise from each of them.

2. Multiply the difference of the two numbers, found by trial, by the least error, and divide the product by the difference of the errors, when they are alike, but by their sum, when they are unlike. Or say, as the difference or sum of the errors is to the difference of the two numbers, so is the least error to the correction of its supposed number.

3. Add the quotient last found to the number belonging to the least error, when that number is too little, but subtract it, when too great; and the result will give the true root nearly.

4. Take this root and the nearest of the two former, or any other, that may be found nearer; and, by proceeding in like manner as above, a root will be had still nearer than before; and so on, to any degree of exactness required. Each new operation commonly doubles the number of true figures in the root.

NOTE I. It is best to employ always two assumed numbers, that shall differ from each other only by unity in the last figure on the right hand; because then the difference, or multiplier, is only 1.

EXAMPLES.

* This rule may be used for solving the questions of Double Position, as well as that given in the Arithmetic, and is preferable for the present purpose. Its truth is easily deduced from the same supposition.

For, by the supposition, rs: x-ax-b, therefore, by division, r-s: s :: b—a : x—b; which is the rule.

EXAMPLES.

1. To find the root of the cubic equation x3+x+x 100, or the value of x in it.

Here it is soon found, that lies between 4 and 5. Assume, therefore, these two numbers, and the operation will be as follows:

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The sum of which is 71.

Then, as 7! : I :: 16: 225.

Hence 4225 nearly.

Again, suppose 42 and 43, and repeat the work as

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Again, suppose 4°264, and 4'265, and work as follows:

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The sum of which is 064087.

Then, as 064087 : 001 :: 4264

To this adding

0'0004299

4'264

We have x very nearly 4 2644299

2. To find the root of the equation x3-15x2+63 go, or the value of x in it.

Here it soon appears, that x is very little above 1.

Suppose, therefore, 1'o and 1'1, and work as follows:

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3'481 sum of the errors.

As 3'481; I :: 1 : 029 correct.

1.00

Hence 1029 nearly.

Again,

Again, suppose the two numbers 103 and 102, and

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NOTE 2. Every equation has as many roots as it con tains dimensions, or as there are units in the index of its highest power. That is, a simple equation has only one value or root; but a quadratic equation has two values or roots; a cubic equation has three roots; a biquadratic equation has four roots, and so on.

And when one of the roots of an equation has been found by approximation, as before, the rest may be found as follows:---Take for a dividend the given equation, with the known term transposed, its sign being changed, to the unknown side of the equation; and for a divisor take x minus the root just found. Divide the said dividend by the divisor, and the quotient will be the equation depressed a degree lower than the given one.

Find a root of this new equation by approximation, as before, and it will be a second root of the original equation. Then, by means of this root, depress the second

equation

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