INVOLUTION.

INVOLUTION is the raising of Powers from any given number,"

as a root.

A Power is a quantity produced by multiplying any given number, called the Root, a certain number of times continually by itself.

Thus,

2=

2 X 2 =

=

2 X 2 X 2 2 × 2 × 2 × 2 =

2 is the root, or 1st power of 2.
4 is the 2d power, or square of 2.
8 is the 3d power, or cube of 2.
16 is the 4th power of 2, &c.

And in this manner may be calculated the following Table of of the first nine powers of the first 9 numbers.

5 25 125 625| 3125 | 15625 78125 390625 1953125 6 36 216 1296 7776 | 46656279936 1679616 10077696

7 49 343 240116807 117649 823543 5764801 40353607

8 64 512 4096 32768 262144 2097152 16777216 134217728 981 729 6561 59049 531441 4782969 430467211387420489|

The

The Index or Exponent of a Power, is the number denoting the height or degree of that power; and it is 1 more than the number of multiplications used in producing the same. So 1 is the index or exponent of the first power or - root, two of the 2d power or square, 3 of the third power or cube, 4 of the 4th power, and so on.

Powers, that are to be raised, are usually denoted by placing the index above the root or first power.

When two or more powers are multiplied together, their product is that power whose index is the sum of the exponents of the factors or powers multiplied. Or the multiplication of the powers, answers to the addition of the indices. Thus, in the following powers of 2,

4. What is the 5th power of 029 ? Ans. •000000020511149.

EVOLUTION.

EVOLUTION, or the reverse of Involution, is the extracting or finding the roots of any given powers.

The root of any number, or power, is such a number, as being multiplied into itself a certain number of times, will produce that power. Thus, 2 is the square root or 2d root of 4, because 22 = 2 x 2 = 4; and 3 is the cube root or 3d root of 27, because 33 = 3 × 3 × 3 = 27.

Any power of a given number or root may be found exactly, namely, by multiplying the number continually into itself. But there are many numbers of which a proposed root can never be exactly found. Yet, by means of decimals, we may approximate or approach towards the root, to any degree of exactness.

Those roots which only approximate, are called Surd roots; but those which can be found quite exact, are called Rational Roots. Thus, the square root of 3 is a surd root; but the square root of 4 is a rational root, being equal to 2: also the cube root of 8 is rational, being equal to 2; but the cube root of 9 is surd or irrational.

Roots are sometimes denoted by writing the character ✔ before the power, with the index of the root against it. Thus, the 3d root of 20 is expressed by 20; and the square root or 2d root of it is 20, the index 2 being always omitted, when only the square root is designed.

When the power is expressed by several numbers, with the sign+or between them, a line is drawn from the top of the sign over all the parts of it: thus the third root of 45 12 is 45-12, or thus

numbers in parentheses.

(45-12), inclosing the

But all roots are now often designed like powers, with fractional indices: thus, the square root of 8 is 8, the cube root of 25 is 253, and the 4th root of 45-18 is 45 — 18), or (45 — 18).

ΤΟ

TO EXTRACT THE SQUARE ROOT.

* DIVIDE the given number into periods of two figures another over each, by setting a point over the place of units, the place of hundreds, and so on, over every second figure, both to the left-hand in integers, and to the right in decimals.

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

* The reason for separating the figures of the dividend into periods or portions of two places each, is, that the square of any single figure never consists of more than two places; the square of a number of So that there two figures, of not more than four places, and so on.

will be as many figures in the root as the given number contains periods so divided or parted off.

And the reason of the several steps in the operation appears from the algebraic form of the square of any number of terms, whether two or three or more. Thus,

(a + b)2=a2 + 2ab + b2=a2 + (2a + b) b, the square of two terms; where it appears that a is the first term of the root, and b the second term; also a the first divisor, and the new divisor is 2a+b, or double the first term increased by the second. And hence the manner of extraction is thus :

1st divisor a) a2 + 2ab + b2 (a + b the root.

Again, for a root of three parts, a, b, c, thus :

(a + b + c) a

a2+2ab+b2 + 2ac + 2bc + c2 = a2 + (2a + b) b + (2a + 26 + c) c, the square of three terms, where a is the first term of the root b, the second, and c the third term; also a the first divisor, 2a + b the second, and 2a +26+c the third, each consisting of the double of And the mode of the root increased by the next term of the same. extraction is thus :

1st divisor a) a2 + 2ab + b2 + 2ac+2bc + c2 (a + b + c the root.

Subtract the square thus found from the said period, and to the remainder annex the two figures of the next following period, for a dividend.

Double the root above mentioned for a divisor; and find how often it is contained in the said dividend, exclusive of its right-hand figure; and set that quotient figure both in the quotient and divisor.

Multiply the whole augmented divisor by this last quotient figure, and subtract the product from the said dividend, bringing down to it the next period of the given number, for a new dividend.

Repeat the same process over again, viz. find another new divisor, by doubling all the figures now found in the root; from which, and the last dividend, find the next figure of the root as before; and so on through all the periods, to the last.

Note, The best way of doubling the root, to form the new divisors, is by adding the last figure always to the last divisor, as appears in the following examples-Also, after the figures belonging to the given number are all exhausted, the operation may be continued into decimals at pleasure, by adding any number of periods of ciphers, two in each period.

NOTE, When the root is to be extracted to many places of figures, the work may be considerably shortened, thus:

Having proceeded in the extraction after the common method, till there be found half the required number of figures