over the place of hundreds, and so on over every second figure, both to the left hand in integers, and to the right hand in decimals, which points will show the number of figures the root will consist of.

2. Find the greatest square number 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 in division,) for the first figure of the root, and the square number, under the period, and substract it therefrom; and to the remainder bring down the two figures of the next following period for a dividend.

3. Place the double o the root, already found, on the left hand of the dividend for a divisor. Seek how often the divisor is contained in the dividend, (excepting the right hand figure,) and place the figure in the root for the second figure of it, and, likewise, on the right hand of the divisor; multiply the divisor, with the last figure annexed, by the last placed in the root, and substract the product from the dividend; to the remainder join the next period for a new dividend.

4. Double the figures already found in the root for a new divisor, (or bring down your last divisor for a new one, doubling the right hand figure of it,) and from these find the next figure in the root, as last directed, and continue the operation in the same manner, till you have brought down all the periods.

EXAMPLES.

1. What is the square root of 625? Ans. 25.

EXPLANATIONS.

625 (25 4

In this example, the greatest square in the left hand period, 6, is 4, of which the root is 2, which must be placed in the quotient, and its square, 4, must be substracted from the period, 6, and to the remainder, 2, the next period, 25, must be brought 45) 225 down, making 225. You must then double the root, 2, and place the double, 4, at the left hand of the divisor, and you will find, that 4 is con

225

tained in 22, the two left hand figures, 5 times; and you must place it, the 5, both in the root, the quotient, and in the divisor; and then you must proceed as in Simple Division, and you will find the quotient, or root, 25, and no remainder, and then the work is done.

2. What is the square root of 729?
3. What is the square root of 1296?
4. What is the square root of 106929?

Ans. 27.
Ans. 36.

Ans. 327.

5. What is the square root of 10342656?

Ans. 3216.

6. What is the square root of 6,9169? Ans. 2,63. 7. What is the square root of,001296? Ans. ,036.

NOTE. The root of a vulgar fraction is found by reducing it to its lowest terms, and extracting the root of the numerator for a new numerator, and of the denominator for a new denominator. If the fraction be a surd, reduce it to a decimal and extract its root.

49

Ans. 2.
Ans. .

8. What is the square root of 4? 9. What is the square root of 162 10. What is the square root of 201? 11. What is the square root of 2 2204? Ans. 4. 12. What is the square root of 42

4225

Ans. 41.

Ans.,7745.

APPLICATION OF THE SQUARE ROOT.

1. A general has an army of 567009 men, how many must he place in rank and file to form them into a square? Ans. 753. 2. Bonaparte's army consisted of 430000 men; when brought into a square how many stood in front? Ans. 700.

3. If the area of a circle be 1521, what is the side of a square equal in area thereto? Ans. 39.

4. A square pavement contains 24336 square stones of equal size; how many are contained in one of its sides? Ans. 156. 5. If 1369 fruit-trees be planted in a square orchard, how many must be in a row each way?

Ans. 37.

6. A square field contains 2025 square rods; how many rods does it measure on each side? Ans. 45.

NOTE. To find a mean proportional between two numbers, you must multiply the given numbers together, and extract the square root of the product.

7. What is the mean proportional between 24 and 96? Ans. 48. 8. What is the mean proportional between 49 and 64? Ans.56. NOTE. The area of a circle is in proportion to the square of its diameter: multiply the square of the diameter by the given ratio, and the square root of the product will be the answer.

9. If the diameter of a circle be 12 feet, what is the diameter of one as large? Ans. ft.

10. A gentleman has two circular ponds in his pleasure ground; the diameter of the less is 100 feet, and the greater is three times as large; what is its diameter ? Ans. 173,2.

NOTE-The square of the hypotenuse, or the longest side of a right angled triangle, is equal to the sum of the squares of the other two sides; and, consequently, the difference of the squares of the hypotenuse and either of the other sides, is the square of the remaining side.

11. What is the length of a ladder that will reach from the top of a wall 32. high, to the opposite side of a ditch 24ft. wide? Ans. 40ft.

12. The length of the rafters of a certain building is 20ft., and the roof is raised in the centre 12ft.; what is the breadth of the building? Ans. 32ft.

13. If a man travel 40 miles due north, and then turn and travel 30 miles due west, how far will he be from the place from which he first started? Ans. 50m.

14. What is the distance between two opposite corners of a field 800 rods long, and 600 rods wide? Ans. 1000 rods.

EXTRACTION OF THE CUBE ROOT.

Q. What is Extraction of the Cube Root?

A. Extraction of the cube root is to find a number, which, being multiplied into its square, will produce the given number.

EXPLANATIONS.

A CUBE is a solid body, having six equal sides, and each of the sides an exact square. The ROOT is, therefore, the measure in length of one of its sides; for, the length, breadth, and thickness of such a cube, body, or square solid, are all alike or equal; and, consequently, the length or root of one side of a cube raised to the 3d power, gives the solid contents.

EXAMPLES.

1. How many solid feet are there in a cubick block, each side measuring 5ft.? Ans. 125ft.

2. How many solid feet are there in a cubick block, each side measuring 8ft.? Ans. 512ft.

RULE.

Q. How do you extract the CUBE ROOT of any given number? A. 1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure from the place of units to the left; and, if there be decimals, to the right.

2. Find the greatest cube in the left hand period, and put its root in the quotient.

3 Substract the cube, thus found, from the said period, and to the remainder bring down the next period, and call this the dividend.

4. Multiply the square of the quotient by 300, calling it the divisor.

5. Seek how many times the divisor may be had in the dividend, and place the result in the quotient or root; then multiply the divisor by this quotient figure, and write the product under the dividend.

6. Multiply the square of this quotient figure by the former figure or figures of the root, and this product by 30, and place the product under the last; under all, write the cube of this quotient figure, and call the amount the subtrahend.

7. Substract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before; and so on, till the whole is finished.

NOTE.-If the divisor can not be had in the dividend, put a cipher in the quotient or root, bring down the next period, and proceed as before directed.

EXAMPLES.

1. What is the cube root of 13824? Ans. 24.

In this example, you first point off the sum into periods of 3 figures each, agreeably to the rule, beginning at the unit's place; then seek the greatest cube in the left hand period, 13, which by the table, you will find is 8, and the root 2; you place the 2, as a quotient, and sub

stract the cube, 8, from the first period, 13; you then bring down the next period, 824, and annex it to the remainder, 5, and call it the dividend: you then multiply the square of the quotient, 2, by 300, for a divisor, which makes 1200; you then seek how many times the divisor is contained in the dividend, which is 4 times; you then multiply the divisor by it, and place the product, 4800, under the dividend; then multiply the square of the last quotient figure, 4, by the quotient obtained before, 2, and that

product by 30, and then place this under the other; and then place the cube of the last quotient figure, 4, under the other two, and add the whole together for a subtrahend, which, substracted from the dividend leaves 0, which shows 24 to be the exact cube of 13824.

2. What is the cube root of 39304? Ans. 34. 3. What is the cube root of 941192? Ans. 98. 4. What is the cube root of 22069810125?

Ans. 2805.

5. What is the cube root of,032768? Ans. 32. 6. What is the cube root of 12,977875? Ans. 2,35. NOTE.-If the root be a surd, reduce it to a decimal before

its root is extracted, as in the square root.

7. What is the cube root of 250?

686

8. What is the cube root of?

Ans.

Ans. 2.

9. What is the cube root of 1331? Ans. 11.

1728

APPLICATION OF THE CUBE ROOT.

NOTE. The sides of cubes are as the cube roots of their solid contents; and, therefore, their contents are as the cubes of their sides. The same proportion is true of the similar sides, or of the diameters of all solid figures of similar forms. EXAMPLES.

1. If a ball, weighing 4lb., be 3in. in diameter, what will be the diameter of a ball, of the same metal, weighing 32lb.? Ans. 6in. 2. The contents of a piece of cubical timber is 103823 solid inches; how many inches is it each way? Ans. 47in.

3. There is a cistern of a cubical form, which contains 1331 cubical feet; what are the length, breadth, and depth of it? Ans. 11ft.

4. How many solid feet of earth must be taken out in digging a cellar, that shall be 12 feet in length, breadth, and depth? Ans. 1728ft.

5. If the diameter of the sun be 112 times that of the earth, how many globes like the earth would it take to make one as large as the sun? Ans. 1404928.

6. The statute bushel contains 2150,4252 cubick or solid inches, what must be the side of a cubick box that shall contain the same quantity? Ans. 12,907in.

7. If a globe of silver, 4 inches in diameter, be worth $150, what is the value of a globe 8in. in diameter? Ans. $1200.