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 288 ft. long, 216 ft. broad, 48 ft. high.

equal to one 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?

298. 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 thei 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.

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?

ART. 300. In every arithmetical series, five things are considered:

1st, the first term; 21, the last term; 3d, the common difference; 4th, the number of terms; 5th, the sum of all the terms.

CASE I.

ART. 301. To find the LAST TERM, when the first term, the common difference, and the number of terms are given.

1. I bought 10 yd. of muslin, at 3 cts. for the 1st yd., 7 cts. for the 2d, 11 cts. for the 3d, and so on, with а com. difference of 4cts.: what did the last yd. cost?

SOLUTION. To find the cost of the second yard, add 4 cts. once to the cost of the first; to find the cost of the third, add 4 cts. twice to the cost of the first; to find the cost of the fourth, add 4 cts. three times to the cost of the first, and so on. Hence,

To find the cost of the tenih yard, add 4cts. nine times to the cost of the first; but 9 times 4 cts. are 36ets., and 3 cts.+36 cts. 39 cts., the cost of the last yard, or last term of the progression.

2. The first term of a decreasing series is 39; the com. diff. 4; the number of terms 10: find the last term.

SOLUTION. In this case, 4 must be subtracted 9 times from 39, which will give three for the last term. Hence, the

Rule for Case I.-Multiply the common difference by the number of terms less one; if an increasing series, add the product to the 1st term: if a decreasing series, subtract the product from the 1st term: the result will be the required term.

3. Find the last term of an increasing progression: the first term 2; the common difference 3; and the number of terms 50. Ans. 149.

4. I bought 100 yd. muslin, at 9 cts. for the 1st yard., 14 cts. for the 2d, and so on, increasing by the com. difference 5 cts. find the cost of the last yd. Ans. $5.04

5. What is the 54th term of a decreasing series, the 1st term 140, and com. diff. 2? Ans. 34. 6. A lends $200 at simple interest, at 8% per annum:

REVIEW.-800. What five things are considered in every series?

at the end of the 1st year $216 will be due; at the end of the 2d year, $232, and so on: what sum will be due at the end of 20 years?

Ans. $520.

7. What is the 99th term of a decreasing series, the 1st term 329, and com diff. ? Ans. 2431.

CASE II.

ART. 302. To find the COMMON DIFFERENCE, when the extremes and the number of terms are given.

1. The first term of a series is 2, the last 20, and the number of terms 7: what the com. diff.?

SOLUTION. The difference of the first and last terms is always equal to the com. diff. multiplied by the number of terms less one (Art. 301); therefore,

If the difference of the extremes be divided by the number of terms less one, the quotient will be the com. diff. Hence, the

Rule for Case II.—Divide the difference of the extremes by the number of terms less one; the quotient will be the com. diff. 2. The extremes are 3 and 300; the number of terms 10: find the com. diff. Ans. 33.

3. A travels from Boston to Bangor in 10 da.; he goes 5 mi. the first day, and increases the distance traveled each day by the same number of miles; and on the last day he goes 50 mi.: find the daily increase. Ans. 5 mi.

ART. 303. It is obvious that if the difference of the extremes be divided by the com. diff., the quotient, increased by unity (1), will be the number of terms.

1. The extremes are 5 and 49; the com. diff. 4: find number of terms. Ans. 12.

CASE III.

ART. 304. To find the SUM of all the terms of the series, when the extremes and number of terms are given.

REVIEW.-301. What is Case 1? What is the Rule for Case 1? 802. What is Case 2? What is the Rule for Case 2?

1. Find the sum of 6 terms of the series whose first term is 1, and last term 11.

Since the two series are the same, their sum is twice the first series. But their sum is obviously as many times 12, (the sum of the extremes), as there are terms. Hence, the

Rule for Case III.—Multiply the sum of the extremes by the number of terms; half the product will be the sum of the series.

2. The extremes are 2 and 50; the number of terms, 24: find the sum of the series. Ans. 624. 3. How many strokes does the hammer of a clock strike in 12 hours?

Ans. 78. 4. Find the sum of the first ten thousand numbers in the series, 1, 2, 3, 4, 5, &c. Ans. 50005000.

5. Place 100 apples in a right line, 3 yd. from each other, the first, 3 yd. from a basket: what distance will a boy travel who gathers them singly and places them in the basket? Ans. 17 mi, 380 yd. 6. A traveled one day 30 mi., and each succeeding day a quarter of a mile less than on the preceding day: how far did he travel in 30 days? Ans. 7914 mi.

7. A body falling by its own weight, if not resisted by the air, would descend in the 1st second a space of 16 ft. I in.; the next second, 3 times that space; the 3d, 5 times that space; the 4th, 7 times, &c.: at that rate, through what space would it fall in 1 min. ? Ans. 57900 ft.

XXVI. GEOMETRICAL PROGRESSION. ART. 305. A Geometrical Progression, or Series, is a series of numbers increasing by a common multiplier, or decreasing by a common divisor. Thus,

1, 3, 9, 27, 48, 24, 12, 6,

81, is an increasing geometric series.

3, is a decreasing geometric series.

REVIEW.-303. How find the number of terms, when the extremes and common difference are given?