920 2. Find a fourth proportional to }, 38, 10. 3. x 10 23 No. required = - X 10 X 8= 6 2 = 306 3 3 VIN. TO FIND A THIRD PROPORTIONAL TO TWO GIVEN NUMBERS. Rule. Multiply the 2nd by itself, and divide by the 1st; the quotient is the number required. EXAMPLES. 867 Find a third proportional to 25, 30: also to 33, 41. 30 X 30 (1) No. required = = 6 X 6 = 36. 25 41 X 41 17 X 17 X 3 (2) No. required 3} 4 X 4 X 10 67 = 5-, 160 = 160 Def. 1. When a number is multiplied by itself, it is said to be involved. Def. 2. The product of a number inultiplied by itself is called a "power" of the number. Def. 3. If the number be numbered by itself once, the power is called the “square," or 2nd power ; if twice, the power is called the “cube,” or 3rd power; and the powers are called the 4th, 5th, &c. according as there are 4, 5, &c. factors, which enter into the composition of the product. TO INVOLVE A NUMBER TO A GIVEN POWER. Rule-1. To square a number, multiply it by itself. 3. If the given power be an odd one, raise the number to the next greater power than the half, and multiply this by the next less. For example, to raise a number to the 5th power, cube it, and multiply the cube by the square. 4. If the given power be an even one, raise the number to a power equal to half the given power, and square it. For example, to raise a number to the 8th power, raise it to the 4th power and square. Def. 1. The process of finding what number, raised to a given power, will produce a given number, is called Evolution. Def. 2. The number which, raised to a given power, will produce a given number, is called the . Root' of the given number, and is styled the square root, cube root, 4th, 5th, &c, root, according as it has to be raised to the 2nd, 3rd, 4th, 5th, &c. power to produce the number. Def. 3. A root is represented by the sign V. Thus V9 = square root of 9; 3127 = cube root of 27, &c. 1. TO FIND THE SQUARE ROOT OF A GIVEN NUMBER. Rule-1. Divide the number into periods of two figures each, by placing a dot over every alternate figure, beginning with the units' place. 2. Find the largest number, whose square does not exceed the first period. This is the 1st figure of the required root. 3. Subtract the square of this from the 1st period. 4. To the remainder annex the 2nd period; call this number the 1st dividend. 5. Double the 1st figure of the root, and write to the left of the 1st dividend, placing a line between them: call this the 1st trial-divisor. 6. Divide the 1st dividend, with the last figure cut off, by the 1st trialdivisor. 7. Write the quotient as the 2nd figure of the root, and also to the right of the first trial-divisor: call this number the 1st divisor. 8. Multiply the 1st divisor by the 2nd figure of the root, and subtract the product from the 1st dividend. 9. To the remainder annex the 3rd period, forming the 2nd dividend. 10. To the last divisor add the last figure found in the root, and place it to the left of the last dividend, thus forming the 2nd trial-divisor. 11. Proceed in the same manner as directed in 6,7, &c. and repeat the process, till all the periods have been used, when, if there is no remainder, the square root will have been found; but if there be one, there is no square root. 12. Whenever any quotient, obtained as directed, gives a subtrahend greater than the dividend, a less number must be taken. 13. This Rule applies to decimals as well as to whole numbers; only in decimals, the number of decimal places must be even, and the root must be made to have half as many as the number. 14. The square root of a fraction is found by extracting the square root of numerator and denominator; or by reducing the fraction to a decimal, and then extracting the root. 15. In finding the square root of a number to a certain number of decimal places, when more than half the figures have been found, the remaining figures may be found by dividing the dividend with the last figure cut off by the last trial-divisor. EXAMPLES. 1. Extract the square root of 6789265609. 6789265609(82397 3. Extract the square root of 141. | 121 V121 11 144 12 4. Extract the square root of 10 to 8 places of decimals. 10(3.16227766 v 144 II. TO EXTRACT THE CUBE ROOT OF A GIVEN NUMBER. Rule-1. Divide the number into periods of three figures, beginning with the units' place, by placing a dot over every third figure. 2. Find the largest number, whose cube does not exceed the first period. 3. Write this as the first figure in the cube root, and subtract its cube from the first period. 4. To the remainder annex the next period; call the number so formed the Ist dividend. 5. Treble the first figure of the root, and write the result some little distance to the left of the dividend; multiply it by the first figure, and write the product nearer the dividend, leaving room for two figures between them. Call this last number the 1st trial-divisor. 6. Divide the dividend, with the last two figures cut off, by the trialdivisor; and write the quotient as the 2nd figure in the root, and also to the right of three times the 1st figure. Call the number so formed, the Ist multiplicand. 7. Multiply the 1st multiplicand by the 2nd figure in the root, and one, write the product under the 1st trial-divisor, keeping two figures to the right; and add the two together; call the sum the 1st divisor. 8. Multiply the 1st divisor by the 2nd figure of the root, and subtract the product from the 1st dividend. 9. To the remainder annex the next period, forming the 2nd dividend. 10. Treble the figures of the root, or add twice the last figure to last multiplicand, and write the product as in 5; add the 1st divisor to the line above it, taking in the square of the last figure of the root; thus forming the 2nd trial-divisor. 11. Proceed in the same manner, as directed in 6, 7, &c. and repeat the process till all the periods have been used, when, if there is no remainder, the cube root will have been formed; but if there be there is no cube root. 12. Whenever any quotient obtained as directed, gives a subtrahend greater than the dividend, a less number must be taken. 13. This rule applies to decimals as well as to whole numbers ; only in decimals, the number of decimal places must be a multiple of 3, and the root will contain one third as many decimal places as the number. 14. The cube root of a fraction may be obtained by extracting the cube root of numerator and denominator; or by reducing the fraction to a decimal, and then extracting the cube root. 15. In finding the cube root of a number to a certain number of decimal places, when one more than half the number of figures required in the root are found, the remaining figures may be found by dividing by the trial-divisor the last dividend with the last two figures cut off. EXAMPLES. 1. Extract the cube root of 625026375. 625026375(85 124 48 496 27064 5296 21184 |