Then, as the difference of these results is to the difference of the two assumed numbers, so is the difference between the true result, given by the question, and either of the former, to the correction of the number belonging to the result used; which correction being added to that number when it is too little, or subtracted from it when it is too great, will give the root required, nearly.

And if the number thus determined, and the nearest of the two former, or any other that appears to be more accurate, be now taken as the assumed roots, and the operation be repeated as before, a new value of the unknown quantity will be obtained still more correct than the first; and so on, proceeding in this manner, as far as may be judged necessary. (k)

(4) The above rule for Double Position, which is much more simple and commodious than the one commonly employed for this purpose, is the same as that which was first given at p. 311 of the octavo edition of my Arithmetic, published in 1810.

To this we may farther add, that when one of the roots of an equation has been found, either by this method or the former, the rest may be determined as follows:

Bring all the terms to the left hand side of the equation, and divide the whole expression, so formed, by the difference between the unknown quantity (x) and the root first found; and the resulting equation will then be depressed a degree lower than the given one.

Find a root of this new equation, by approximation, as in the first instance, and the number so obtained will be a second root of the original equation.

Then, by means of this root, and the unknown quantity, depress the second equation a degree lower, and thence find a third root; and so on, till the equation is reduced to a quadratic; when the two roots of this, together with the former, will be the roots of the equation required.

Thus, in the equation x3-15x2 +63x=50, the first root is found, by approximation, to be 1.02804. Hence

x—1.02804 (‍x3 — 15x2 + 63x – 50(x2 – 13 97196x+48.63627=0 And the two roots of the quadratic equation, x2-13 97196x= — 48.63627, found in the usual way, are 6.57653 and 7.39543,

EXAMPLES.

1. Given x3+x2+x=100, to find an approximate value of x.

Here it is soon found, by a few trials, that the value of a lies between 4 and 5

Hence, by taking these as the two assumed numbers, the operation will stand as follows:

First Sup.

Second Sup.

And consequently x=4+ 225-4225, nearly. Again, if 1.2 and 4.3 be taken as the two assumed numbers, the operation will stand thus:

So that the three roots of the given cubic equation x3 - 15x2 +63x=50, are 1.02804, 6.57653, and 7.39543; their sum being 15, the coefficient of the second term of the equation, as it ought to be when they are right.

And consequently x=4.3-.036-4.264, nearly. Again, let 4.264 and 4.265 be the two assumed numbers; then

2. Given (x2-15)2+x√x=90, to find an approximate value of x.

Here, by a few trials, it will be soon found, that the value of x lies between 10 and 11; which let, therefore, be the two assumed numbers, agreeably to the directions given in the rule.

31.122: .482

64.5 : 1 ::

And consequently x= 11-.482 10.518

Again, let 10.5 and 10.6 be the two assumed numbers

1. Given x2+10x2+5x=2600, to find a near approximate value of x. Ans. 11.00673

2. Given 2x4 - 16x3 +40x3-30x+1=0, to find a near value of x. Ans. x 1.284724

3. Given x+2x4+3x3+4x2+5x=54321, to find the value of x. Ans. 8414455

4. Given /7x+4x3 +√20x3 — 10x=28, to find the value of x. Ans. 4510661

5. Given 144x2 - (x2+20)2 +✔✓196x2 — (x2+24)3· 114, to find the value of x. Ans. 7.123883

OF EXPONENTIAL EQUATIONS.

AN exponential quantity is that which is to be raised to some unknown power, or which has a variable quantity for its index; as

And an exponential equation is that which is formed between any expression of this kind and some other quantity, whose value is known; as

a*=b, xx=a, &c.

Where it is to be observed, that the first of these equations, when converted into logarithms, is the same as

x log. a=b, or x=

log. b
log. a

; and the second equation xα,

is the same as x log. x=log. a.

In the latter of which cases, the value of the unknown quantity may be determined, to any degree of exactness, by the method of double position, as follows:

RULE.

Find, by trial, as in the rule before laid down, two numbers as near the number sought as possible, and substitute them in the given equation.

x log. x= log. a,

instead of the unknown quantity, noting the results obtained from each.

Then, as the difference of these results is to the difference of the two assumed numbers, so is the difference between the true result, given in the question, and either of the former, to the correction of the number belonging to the result used; which correction being added to that number, when it is too little, or subtracted from it, when it is too great, will give the root required, nearly.

And, if the number, thus determined, and the nearest of the two former, or any other that appears to be nearer, be taken as the assumed roots and the operation be repeated as before, a new value of the unknown quantity will be obtained still more correct than the first; and so on, proceeding in this manner, as far as may be thought necessary.

EXAMPLES.

1. Given 100, to find an approximate value of x. Here, by the above formula, we have

x log. x log. 100-2.

And since x is readily found, by a few trials, to be