2. What number is that which added to its square shall make the Sum 12?

But the required number is 3, (for 3 added to 9, the square of 3, make 12) therefore the rule fails in this example. The true answer, however, may be approximated to any assigned degree of accuracy by repeating the ope ration, and constantly making the last quotients or approximations, the assumed numbers:

Next, making 3.2 and 3:02 the assumptions we have

3.02 X 3.02 = 9 1204 the square of 3.02

Again, let the suppositions be 3.02 and 3.0005; and the next approxi mation comes out 3:000001. And if the operation be repeated with 3.0005 and 2000001, the result will be 3.00000000002, &c.

In this manner the rule may frequently be applied with success in very difficult cases.

OF INVOLUTION.

110. WHEN a number is multiplied into itself a certain number of times, it is called Involution, or raising of powers.

The number so multiplied is the root; and the products are the powers.

111. The power to which a number is to be raised is usually denoted by a small figure called the index or exponent.

Thus 53 denotes the 3d power or cube of 5.

or exponents of the powers are 3, 4, and 2.

Since 2 × 2 × 2 × 2 × 2 = 32 is the 5th power of the Foot 2, it follows that the 5th power is the product of the square and cube.

For 2×24 is the square; and 2x2x2 = 8 is the cube; therefore 4 × 832 the 5th power.

Hence 2 × 2325; consequently the addition of the indices 2 and 3 answer to the multiplication of powers; viz. 22 × 23 = 22+ 3.

Also 33 x 34 37. For 33 is 27; and 34 is 81; and 27 × 81 is equal to 2187 = 3 × 3 × 3 × 3 × 3 × 3 × 3

2012-01 × 2014.0401 the square, which squared is 4-04014·0401 × 4·040116·32240801, the Answer.

EVOLUTION.

112. EVOLUTION is the extraction or finding the roots of any given powers, being the reverse of Involution.

Every number which is a known power will have a determinate root called a rational root: thus the number 8 is a cube number whose root is 2; and the number 9 is a square having 3 for the root: but 10 is not an exact power of any kind, be cause its root can never be accurately obtained. By the help of decimals however, the roots of any numbers may be approximated to any assigned degree of exactness: these approximate

roots are called irrational or surd roots. will be a surd. And the square root of 8; 9 are both surds,

113.

Rule.

Thus any root of 10 and the cube root of

tract the SQUARE ROOT.

Rule. Begin at the units place and point the number into periods of two figures each.

Find the greatest square in the first period on the left hand and set its root on the right of the given number, in the same manner as a quotient figure in division.

Subtract the square from the period above it, and to the re mainder bring down the next period, for a dividend.

Double the aforesaid root, and find how often it is contained in the dividend, exclusive of its rig hand figure, and set the result in the quotient, and also on the right of the divisor.

Multiply the augmented divisor by this last quotient figure, and subtract the product from the dividend, and to the remainder bring down the next period for a new dividend.

Then find a new divisor by doubling the figures of the quotient; and proceed as before till all the periods are brought down.

The best way of doubling the root or quotient is by adding the last figure always to the last divisor.

Examples.

1. Required the square root of 41409225 ?

41409225 (6435 root or quotient.

36

124) 540

4. 496

1283) 4492

3 3819

12865) 64325

64325

114. The rule for extracting the square root is easily derived from the following method of forming a square or the product of two like numbers. For example, suppose 6433 × 0435 (the above square).

The sum of 64 x 4 and 60 x 4 being the same as 4 multiplied by twice 60 added to 4, or 124 x 4; therefore to find the difference of the squares of 60 and 64, add 4 to twice 60 and multiply the sum by 4.

In like manner the difference of the squares of 640 and 643 will be 3 added to twice 640 and the sum multiplied by 3, (1283 × 3).

And the difference between the squares of 6430 and 6435, is 5 added to twice 6430, and the sum multiplied by 5, or 12865 × 5; and so on.

Hence 6135 x 6435, or the square of 6435 will be

6000 X 6000 = 36000000

Therefore as the whole square consists of the products 6000 × 6000, 12400 X 400, 12830 X 30, and 12865 X 5, if it be divided by 6000, and then the remainder by 12400, and the next remainder by 12830, and the last remainder by 12865, the quotients will be 6000, 400, 30, and 5, whose

is the root.

In this operation the first divisor is the thousands in the root; the second is double the thousands added to the hundreds; the third is double the thousands and hundreds added to the tens; and the fourth is double the