INVOLUTION3.

261. The power of any number is the product that arises by multiplying that number into itself; and the product (if neeessary) into the given number; and this product (if necessary) into the given number, and so on continually.

262. The product arising from one multiplication, is called the square, or second power of the given number; the product arising from two successive multiplications, is called the cube, or third power; the product of three multiplications, the fourth power; of four multiplications, the fifth power, and so on.

263. The power of any number is denoted by a small figure, called the index or exponent of the power, placed over, and a little to the right of, the given number.

Thus 32 denotes the second power, or square of 3, the small 2 being the index or exponent of the second power; 79 denotes the third power of 7, where 3 is the index; 21 denotes the fifth power of 21, where 5 is the index, &c, 264. Involution teaches to find the powers of any given number.

265. To involve whole numbers or decimals to any power. RULE I. Multiply the given number into itself for the square, and this product into the given number for the cube, and so on continually for the higher powers; observing, that to obtain any power, the number of successive multiplications will always be one less than the index of the required power.

II. If there are decimals in the number given to be involved, mark off the decimals in each product, according to the rule for multiplying decimals, Art. 224.

• The name Involution is derived from the Latin involvo, to wrap or fold in, The number to be involved is called the root of the proposed power; the num ber arising from the involution is called the power of the given root. The terms square and cube are applied to certain numbers, because they arise by processes similar to the known method of computing the capacity of those figures: and because the second power is called a square, and the third power a cube, the second root is named the square root, and the third root the cube root; the fourth power is sometimes called the biquadrate, (bis quadratus,) and the fourth root the biquadrate root. Particular names for other powers and roots are to be found in old books, but they are now seldom used; see the note on Art. 52. part 3.

EXAMPLES.

1. What is the fourth power of 12?

OPERATION.

12 1st power.

12

Explanation.

Here the index of the required power being 4, 144 = 2nd power. three multiplications are necessary: the first produces the square, or second power; the second produces the cube, or third power; and the third produces the biquadrate, or fourth power, as was required.

12 1728 3d power.

12

20736 = 4th power.

2. Involve 2.3 and .103 each to the fifth power.

3. Involve 234 to the square. Square .54756. 4. What is the cube of 54 ? Cube 157464.

5. Involve 100.2 to the third power. Third power 1006012.008. 6. Involve 94.75 to the fourth power.

80596628.44140625.

Fourth power

266. To involve a simple fraction to any power.

RULE. Involve the numerator and denominator each séparately to the given power, and the results will be the respective terms of a new fraction, which will be the power required.

267. To involve a mixed number to any power.

RULE I. Either reduce the given mixed number to an improper fraction, by Art. 172; involve both terms of this fraction by the last rule; reduce the resulting improper fraction to its proper terms, by Art. 173, and the result will be the power. Or,

II. Reduce the fractional part of the given number to a decimal, Art. 233. subjoin this to the whole number, and involve the result to the given power, by Art. 265.

11. Involve 2 to the second power.

required, or,

=

= (Art. 173.) 7 the power

Secondly, 22.75 by Art. 233.

Then 2.75)2=2.75 × 2.75 = 7.5625, the power, as before; for this decimal .5625 being reduced to a vulgar fraction, (Art. 232.) will as above.

12. Involve

to the cube.

4%

EVOLUTION t.

268. The root of any number is that which being multiplied once, or oftener continually into itself, will produce the said number.

269. A number which being multiplied once into itself produces the given number, is called the square root of that number; a number which multiplied successively twice, produces the given number, is called the cube root of that number; if it produces the given number by three successive multiplications, it is called the fourth, or biquadrate root of that number, and so on. 270. The root of any number is denoted either by a radical sign, with a small figure expressive of the number of the root

The name Evolution is derived from the Latin evolvo, to unfold. With respect to the operation, the evolution of roots consisting of only one figure is merely a simple mechanical process, the reason of which immediately appears : but when the root consists of several figures, the grounds of the rule by which it is extracted are by no means obvious. A respectable writer, whose name would do honour to these pages, observes, that "any person who can extract the 66 square and cube root in Algebra, will not be at a loss to demonstrate the "rules of square and cube root" in Arithmetic; "and to those who cannot, "a demonstration would be of little or no use." The truth is, that the common rules for the extraction of roots, either in Algebra or Arithmetic, as far as I have been able to learn, have never yet been demonstrated independently, and without supposing that both the root and its power are previously known: having the root given, its power, although unknown, is easily obtained by multiplication; but the root being unknown, cannot be obtained from the power by the converse operation of division, because the divisor is not known: hence it appears, that the rules for Evolution were first discovered mechanically, or by dint of trial; and the only proof that they are true is, that the number arising from their operation, being involved, produces the given power. See the note on Art. 57. part 3. Mr. Wood has shewn very clearly in the extraction of the cube root, how the several steps in the Arithmetical and Algebraic ope rations respectively coincide with each other. Elem. of Algeb. pp. 62, 63. Third Edit.

The square root of any number, is a mean proportional between unity and that number; the cube root is the first of two mean proportionals between unity and the given number; the biquadrate root is the first of three mean proportionals between unity and the given number; and in general, if between unity and any power there be taken mean proportionals in number one less than the index of that power, the first of these will be the root required.

over it, and the whole placed before the given number; or by a fractional index or exponent, placed over, and a little to the right of the given number.

Thus 3 or 3 denotes the square root of 3; 3√7 or 7 denotes the cube root of 7; 5/21 or 21 denotes the fifth root of 21, &c.

271. Evolution teaches to find the roots of any given number.

Thus to extract the square root of a number, is to find such a number which being multiplied once into itself, produces the given number: to extract the cube root, is to find a number which being multiplied into itself, and that product into the same number, produces the given number; and so for other roots.

EXTRACTION OF THE SQUARE ROOT.

272. The following table contains the first nine whole numbers which are exact squares, with the square root of each placed under its respective number.

....

SQUARES.. 1. 4. 9. 16. 25. 36. 49. 64. 81. SQUARE ROOTS. 1. 2. 3. 4. 5. 6. 7. 8. 9.

From this table the root of any exact square, being a single figure, may be obtained by inspection, as is plain.

273. To extract the square root, when it consists of two or more figures, from any number.

RULE I. Make a point over the units' place of the given num ber, another over the hundreds, and so on, putting a point over every second figure; whereby the given number will be divided into periods of two figures each, except the left hand period, which will be either one or two, according as the number of figures in the whole is odd or even.

II. Find in the table the greatest square number not greater than the left hand period, set it under that period, and its root in the quotient.

III. Subtract the said square from the figures above it, and to the remainder bring down the next period for a dividend. IV. Double the root, (or quotient figure,) and place it for a divisor on the left of the dividend.

V. Find how often the divisor is contained in the dividend, omitting the place of units, and place the number (denoting