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Hence the following

RULE.

1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure beyond the place of units.

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

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

4. Multiply the square of the quotient by 300, calling it the triple square; multiply also the quotient by 30, calling it the triple quotient; the sum of these call the divisor.

5. Find how many times the divisor is contained in the dividend, and place the result in the quotient.

6. Multiply the triple square by the last quotient figure, and write the product under the dividend ; multiply the square of the last quotient figure by the triple quotient, and place this product under the last; under all, set the cube of the last quotient figure, and call their sum the subtrahend.

7. Subtract 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 completed.

Note 1. The same rule must be observed for continuing the oper. ation, and pointing for decimals, as in the square root.

Note 2. In inquiring how many times the dividend will contain the divisor, we must sometimes make an allowance of two or three units. See National Arithmetic, page 205. 1. What is the cube root of 78402752 ?

OPERATION.

78402752(428

4x4x300= 64

4x30= 4920) 14402=1st dividend. 1st divisor.= 9600

4800 X2= 480

120 x2x2= 8

2x2x2 10088=1st subtrahend. Ist subtrahend.=

4800

120 4920 9600 480

8 10088

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158

APPLICATION OF THE CUBE ROOT. [Sect. 49.

530460)4314752=2d dividend. 42x42x300= 529200 4233600

42x30= 1260 80640

2d divisor.= 530460 512

529200x8=4233600 4314752—2d subtrahend. 1260x8x8= 80640

8x8x8= 512

2d subtrahend.=4314752 2. What is the cube root of 74088 ?

Ans. 42. 3. What is the cube root of 185193 ?

Ans. 57. 4. What is the cube root of 80621568 ? Ans. 432. 5. What is the cube root of 176558481 ? Ans. 561. 6. What is the cube root of 257259456 ? Ans. 636. 7. What is the cube root of 1860867 ? Ans. 123. 8. What is the cube root of 1879080904 ? Ans. 1234. 9. What is the cube root of 41673648.563 ?

Ans. 346.7. 10. What is the cube root of 48392.1516051 ?

Ans. 78.51. 11. What is the cube root of 8.144865728 ?

Ans. 2.012. 12. What is the cube root of 7296 ?

Ans. 16 13. What is the cube root of 49 ?

Ans. 3. 14. What is the cube root of 1663 ?

Ans. 53. 15. What is the cube root of 855?

Ans. 43.

APPLICATION OF THE CUBE ROOT.

Spheres are to each other, as the cubes of their diameter.

Cones are to each other, as the cubes of their altitudes or bases.

All similar solids are to each other, as the cubes of their homologous sides. 16. If a ball, 4 inches in diameter, weighs 50lbs., what is the weight of a ball 6 inches in diameter ? ,

Ans. 168.7+ lbs. 17. If a sugar loaf, which is 12 inches in height, weighs

16lbs., how many inches may be broken from the base, that the residue may weigh Slbs. ? Ans. 2.5+ in.

18. If an ox, that weighs 800lbs., girts 6 feet, what is the weight of an ox that girts 7 feet ? Ans. 1270.3lbs. 19. If a tree, that is one foot in diameter, make one cord, how many cords are there in a similar tree, whose diameter is two feet ?

Ans. 8 cords. 20. If a bell, 30 inches high, weighs 1000lbs., what is the weight of a bell 40 inches high? Ans. 2370.3lbs. 21. If an apple, 6 inches in circumference, weighs 16 ounces, what is the weight of an apple 12 inches in circumference ?

Ans. 128 ounces.

Section 50.

GEOMETRICAL PROBLEMS.

1. To find the area of a square or parallelogram.

Rule. Multiply the length by the breadth, and the product is the superficial contents.

2. To find the area of a rhombus or rhomboid.

Rule. Multiply the length of the base by the perpendicular height. 3. To find the area of a triangle.

Rule. Multiply the base by half the perpendicular height ; or, add the three sides together ; then take half of that sum, and out of it subtract each side severally; multiply the half of the sum and these remainders together, and the square root of this product will be the area of the triangle.

4. Having the diameter of a circle given, to find the circumference.

Rule. Multiply the diameter by 3.141592, and the product is the circumference.

Note. The exact proportion, which the diameter of a circle bears to the circumference, has never been discovered, although some mathematicians, have carried it to 200 places of decimals. If the diameter of a circle be 1 inch, the circumference will be 3.141592653 5897932384626433832795028941971693993751058209749445923078164062 8620899862803482534211706798214808651328230664709384464609550518 22317253594081284802 inches nearly. 5. Having the diameter of a circle given, to find the side of an equal square.

Rule. Multiply the diameter by .886227, and the product is the side of an equal square. 6. Having the diameter of a circle given, to find the side of an equilateral triangle inscribed.

Rule. Multiply the diameter by .707016, and the product is the side of a triangle inscribed. 7. Having the diameter of a circle given, to find the area.

Rule. Multiply the square of the diameter by .785398, and the product is the area. Or, multiply half the diameter by half the circumference, and the product is the area. 8. Having the circumference of a circle given, to find the diameter.

Rule. Multiply the circumference by .31831, and the product is the diameter. 9. Having the circumference of a circle given, to find the side of an equal square.

Rule. Multiply the circumference by .282094, and the product is the side of an equal square. 10. Having the circumference of a circle given, to find the side of an equilateral triangle inscribed.

Rule. Multiply the circumference by .2756646, and the product is the side of an equilateral triangle inscribed. 11. Having the circumference of a circle given, to find the side of an inscribed square.

Rule. Multiply the circumference by .225079, and the product is the side of a square inscribed.

12. To find the contents of a cube or parallelopipedon.

Rule. Multiply the length, height, and breadth, continually together, and the product is the contents. 13. To find the solidity of a prism.

RULE. Multiply the area of the base, or end, by the height. 14. To find the solidity of a cone or pyramid.

Rule. Multiply the area of the base by 1 of its height. 15. To find the surface of a cone.

Rule. Multiply the circumference of the base by half its slant height. 16. To find the solidity of the frustum of a cone, or pyramid.

RULE. Multiply the diameters of the two bases together, and to the product add of the square of the difference of the diameters; then multiply this sum by .785398, and the product will be the mean area between the two bases ; lastly, multiply the mean area by the length of the frustum, and the product will be the solid contents.

Or, find when it would terminate in a cone, and then find the contents of the part supposed to be added, and take it away from the whole. 17. To find the solidity of a sphere or globe.

RULE. Multiply the cube of the diameter by .5236. 18. To find the convex surface of a sphere or globe.

Rule. Multiply its diameter by its circumference. 19. To find the contents of a spherical segment.

Rule. From three times the diameter of the sphere, take double the height of the segment ; then multiply the re. mainder by the square of the height, and the product by the decimal .5236 for the contents ; or to three times the square of the radius of the segment's base, add the square of its

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