now there are 400 sq. in. in the face of each, and 400 × 3=1200 sq. in. in one face of the three; then multiplying by 4, (the thickness,) gives 4800 cu. in. for their contents.

The solid contents of the three oblong solids, C, c, c, are found (Art. 93) by multiplying the number of sq. in. in the face by the thickness; now there are 20 X 480 sq. in. in one face of each, and 80 × 3=240 sq. in. in one face of the three; then multiplying by 4, (the thickness,) gives 960 cu. in. for their contents.

Lastly, find the contents of the small cube, D, by multiplying its length (4) by its breadth (4), and that product by the thickness (4); this gives 4X4 X 4 = 64 cu. in.

If the solid contents of the several additions be added together, their sum, 5824 cu. in., will be the number of small cubes remaining after forming the first cube, A.

ADDITIONS.

4800 cu. in.

B, B,

B,

=

C, c,

c,= 960

D,

=

64"

Sum 5824

Hence, when 13824 cu. in. are arranged in the form of a cube, each side is 24 in.; that is, the cube root of 18824 is 24.

In finding the solid contents of the additions, in each case the last multiplier is the thickness.

To produce the same result more conveniently, find the area of one face of each of the additional solids, then the sum of the areas, (as in the numerical operation,) and multiply it by the common thickness.

The sum of the areas of one face of each of the additional solids, is termed the COMPLETE DIVISOR. Thus,

In the preceding operation, 1456 is the complete divisor.

NOTE. As the 1st figure of the root is always in the tens' place with regard to the 2d, annex to it a cipher before it is squared; or, omit the cipher, and multiply the square by 300 instead of 3.

REVIEW.-295. Ilow obtain the 1st figure of the root? Why square it ? Why multiply by 3? What is the product called? Why?

295. How cbtain the 2d figure of the root? Why multiply the 1st figure by the 2d? Why multiply their product by 3? Why square the 2d figure of the root? How find the subtrahend? What is the complete divisor?

295. NOTE. Why is a cipher annexed to the 1st figure of the root before squaring? If the cipher is omitted, what must be done? What if the cipher is omitted in multiplying the 1st figure of the root by the 2d?

3d Bk.

19

For the same reason, annex a cipher to the figure first obtained, before multiplying it by the 2d (the thickness):

Or, omit the cipher, and multiply by 30 instead of 3.

Ans. 12.

*2. What is the cube root of 1728?

3. Find the cube root of 413493625.

74 X 74 X 300=1642800, 8269625 = dividend.

74 X 5 X 30= 11100

5 X 5

=

25

1653925

8269625=subtrahend.

EXPLANATION.-By the rule for pointing (Art. 294), the root will contain 3 figures. Find the 1st and 2d figures of the root as in the preceding examples. Then consider 74 as so many tens, and find the 3d figure in the same manner as the 2d was obtained. 4. Find the cube root of 515.849608

EXP.-After obtaining the 1st figure, and bringing down the 2d period, we find the trial divisor is not contained in the dividend;

therefore, place a 0 in the root, and bring down another period. Hence, the cube root of a decimal is found in the same manner as that of a whole number, the periods being reckoned both ways from the decimal point.

ART. 296. TO EXTRACT THE CUBE ROOT,

Rule.-1. Separate the given number into periods of 3 places each, by placing a dot over the units, a dot over the thousands, and so on. (The left period often has only one or two figures.) 2. Find the greatest cube in the left period, and place its root on the right, as in division. Subtract the cube of the root from the left period, and to the remainder bring down the next period for a dividend.

3. Square the root found, and multiply it by 300 for a trial divisor. Find how many times this divisor is contained in the dividend, and write the result in the root. Multiply the last figure of the root by the rest, and by 30; square the last figure of the root, and add these two products to the trial divisor; the sum will be the COMPLETE divisor.

4. Multiply the complete divisor by the last figure of the root, and subtract the product from the dividend; to the remainder bring down the next period for a new dividend, and so proceed until all the periods are brought down.

NOTES.-1. When the product of the complete divisor by the last figure of the root is larger than the dividend, the figure of the root must be diminished.

2. After bringing down all the periods, if there be a remainder, the operation may be continued by annexing periods of ciphers. 3. If the divisor is not contained in the dividend, write a cipher in the root, and bring down another period for a new dividend.

4. When there are decimals in the given number, separate them into periods by placing dots over the tenths, ten-thousandths, and so on. The reasons for this are similar to those in Art. 286, Ex. 4.

5. To extract the cube root of a common fraction, reduce it to its lowest terms; then, if both terms are perfect cubes, extract the cube root of each; but, if either term is an imperfect cube, reduce the fraction to a decimal, and then extract the root.

B EVERY TEACHER should have a set of cubical blocks.

TO TEACHERS.-Instead of finding the subtrahend by the rule, it may be obtained by finding separately the contents of each solid, then adding the whole together. This method, in connection with the blocks, is best adapted to give a clear idea of the nature of the operation.

25. The contents of a cubical cellar are 1953.125 cu.

ft. find the length of one side.

26. In 1 cu. ft., how many 3 in cubes?

Ans. 12.5 ft.

Ans. 64.

27. How many cubical blocks, each side of which is one-quarter of an inch, will fill a cubical box, each side of which is 2 inches?

Ans. 512.

28. Find the difference between half a solid foot, and a solid half foot. Ans. 648 cu. in.

29. Find the side of a cubical mound equal to one 288 ft. long, 216 ft. broad, 48 ft. high. Ans. 144 ft. 30. The side of a cubical vessel is 1 foot: find the side of another cubical vessel that shall contain 3 times as much. Ans. 17.306+in.

REVIEW.-296. What the rule for extracting the cube root? NOTES. When the subtrahend is larger than the dividend, what is required?

296. When there is a remainder, how continue the operation? How proceed when there are decimals in the given number? How extract the cube root of a common fraction?

ART. 297. It is a known principle, that spheres are to each other as the cubes of their diameters; and that

All similar solids are to each other as the cubes of theix corresponding sides.

Hence, the solid contents, or weight of two similar solids, have to each other the same ratio as the cubes of their like parts.

31. A metal ball 6 in. in diameter weighs 32 lb.: what is the weight of one of the same metal, whose diameter is 3 in.? Ans. 4 lb. 32. If the diameter of Jupiter is 11 times that of the earth, how many times larger is it?

Ans. 1331.

ART. 298. THE CUBE ROOT BY FACTORING.

The cube root of any perfect cube may be extracted by resolving the given number into its prime factors, and multiplying together one of each three equal factors.

1. Find the cube root of 216.
2. Find the cube root of 27 × 64.

Ans. 3X2=6.

Ans. 12.

XXV. ARITHMETICAL PROGRESSION. ART. 299. An Arithmetical Progression, or Series, is a series of numbers which increase or decrease, by a common difference. If the series increase, it is called an increasing series; if it decrease, a decreasing series.

Thus, 1, 3, 5, 7, 9, 11, &c., is an increasing series. 20, 17, 14, 11, 8, 5, &c., is a decreasing series.

The numbers forming the series are called terms; the first and last terms are the extremes; the other terms, the means.

REVIEW.-297. What ratio have the solid contents of two similar bodies? 298. How extract the cube root of a perfect cube by factoring? 299. What is an arithmetical progression? When is the series increasing? Decreasing? Give examples. What are the extremes? The means?