: 26. How many solid inches in a cubical block, which measures 1 inch on each side? How many in one measuring 2 inches on each side? 3 inches on each side? 4 inches on each side? 6 inches on each side? 10 inches on each side? 20 inches on each side? A. 1, 8, 27, 64, 216, 1000, 8000. 27. What is the length of one side of a cubical block, which contains 1 solid or cubic inch? 8 solid inches? 27 solid inches? 64 solid inches? 125 solid inches? 216 solid inches? 1000 solid inches? 8000 solid inches? A. 1, 2, 3, 4, 5, 6, 10, 20.

By the two preceding examples we see that the sides of the cube are as the cube roots of their solid contents, and their solid contents as the cubes of their sides. It is likewise true, that the solid contents of all similar figures are in proportion to each other as the cubes of their several sides or diameters.

Note. The relative length of the sides of cubes, when compared with their solid contents, will be best illustrated by reference to the cubical blocks, accompanying this work.

28. If a ball, 3 inches in diameter, weigh 4 pounds, what will a ball of the same metal weigh, whose diameter is 6 inches?

33634: 32: Ratio, 23 X 4 = 32 lbs., Ans. 29. If a globe of silver, 3 inches in diameter, be worth $160, what is the value of one six inches in diameter? 3363 $160 $1280, Ans. 30. There are two little globes; one of them is 1 inch in diameter, and the other 2 inches; how many of the smaller globes will make one of the larger? A. 8.

31. If the diameter of the planet Jupiter is 12 times as much as the diameter of the earth, how many globes of the earth would it take to make one as large as Jupiter? A. 1728.

32. If the sun is 1000000 times as large as the earth, and the earth is 8000 miles in diameter, what is the diameter of the sun? A. 800000 miles Note. The roots of most powers may be found by the square and cupe roots only; thus the square root of the square root is the biquadrate or 4th root, and the 6th root is the cube of this square root.

ARITHMETICAL PROGRESSION.

¶ LXXXVIII. Any rank or series of numbers more than 2, increasing by a constant addition, or decreasing by a constant subtraction of some given number, is called an Arithmetical Series, or Progression.

The number which is added or subtracted continually is called the common difference.

When the series is formed by a continual addition of the common difference, it is called an ascending series; thus,

2, 4, 6, 8, 10, &c., is an ascending arithmetical series; but

10, 8, 6, 4, 2, &c., is called a descending arithmetical scries, because it is formed by a continual subtraction of the common difference, 2.

The numbers which form the series are called the terms of the series or progression. The first and last terms are called the extremes, and the othe: terms the means.

In Arithmetical Progression there are reckoned 5 terms, any three of which being given, the remaining two may be found, viz.

The first term, the last term, and the number of terms, being given, to find the common difference;-

1. A man had 6 sons, whose acveral ages differed alike; the youngest was 3 rears old, and the oldest 28; what was the common difference of their ages The difference between the youngest son and the eldest evidertly shows tho increase of the 3 years by all the subsequent additions, till we come to 28 years; and, as the number of theso additions are, of course, 1 less than the number of sons, (5), it follows, that, if we divide the whole difference (28-3), 25, by the number of additions, (5), wo shall have the difference between each ono separately, that is, the common difference.

Thus, 28-325; then, 255-5 years, the common difference. A.5 yrs Hence, to find the common difference ;

Divide the difference of the extremes by the number of terms, ess 1, and the quotient will be the common difference.

2. If the extremes bo 3 and 23, and the nuniber of terms 11, what is the common difference? A. 2.

3. A man is to travel from Boston to a certain place in 6 days, and to go only 5 miles the first day, increasing the distance travelled each day by un equal excess, so that the last day's journey may be 45 miles; what is the daily increase, that is, the common difference? A. 8 miles.

4. If the amount of $1 for 20 years, at simple interest, be $2,20, what is the rate per cent.?

In this example we see the amount of the first year is $1,06, and the last year $2,20; consequently, tho extremes are 106 and 220, and the number of terms 20. A. $,066 per cent.

5. A man bought 60 yards of cloth, giving 5 cents for the first yard, 7 for the second, 3 for the third, and so on to the last; what did the last cost?

Since, in this example, we have the common difference given, it will be easy to find the price of the last yard; for, as there are as many additions as there are yards, less 1, that is, 59 additions of 2 cents, to be made to the first yard, it follows, that the last yard will cost 259 118 cents more than the first, and the whole cost of the last, reckoning the cost of the first yard, will be 1185 $1,23. A. $1,23.

Hence, when the common difference, the first term, and the number of terms, are given, to find the last term;

Multiply the common difference by the number of terms, less 1, and add the first term to the product.

6. If the first term be 3, the common difference 2, and the number of tering Il, what is the last term? A. 23.

7. A man, in travelling from Boston to a certain pace in 6 days, travelled the first day 5 miles, the second 8 miles, travelling each successive day 3 miles farther than the former; what was the distance travelled the last day? A. 20. 8. What will $1, at 6 per eca, amount to, in 20 years, at simple interest? The common difference is the 6 per cent.; for the amount of $1, for 1 year, is $1,06, and $1,06 + $,06 = $1,12 the second year, and so on. A. $9,20. 9. A man bought 10 yards of cloth, in arithmetical progression; for the 1st var he gave 6 cents, and for the last yard he gave 24 cents; what was the amount of the whole ?

In this example it is plain that the cost of the first and last yards wil be the averago price of the whole number of yards; thus, 6 cts. + 24 cts. 30 215 cts., average price; then, 10 yds. X 15 150 cts. $1,50, who le

cost

A. $1,59.

Hence, when the extremes, and the number of terms are given, to find the sum of all the terms;

Multiply the sum of the extremes by the number of terms, and the product will be the answer.

10. If the extremes be 3 and 273, and the number of terms 40, what is the sum of all the terms? A. 5520.

11. How many times does a regular clock strike in 12 hours? A. 78.

12. A butcher bought 100 oxen, and gave for the first ox $1, for the second $2, for the thira $3, and so on to the last; how much did they come to at that rate? A. $5050.

13. What is the sum of the first 1000 numbers, beginning with their natura order, 1, 2, 3, &c.? A. 500500.

14. If a board, 18 feet long, be 2 feet wide at one end, and come to a point at the other what are the square contents of the board? A. 18 feet.

15. If a piece of land, 60 rods in length, be 20 rods wide at one end, and a the other terminate in an angle or point, what number of square rods docs i contain? A. 600.

16. A person, travelling into the country, went 3 miles the first day, and increased every day's travel 5 miles, till at last he went 58 miles in one day; how many days did he travel?

We found, in example 1, the difference of the extremes, divided by the number of terms, less 1, gave the common difference; consequently, if, in this example, we divide (583) 55, the difference of the extremes, by the common difference, 5, the quotient, 11, will be the number of terms, less 1; then, 1+1 12, the number of terms. A. 12.

Hence, when the extremes and common difference are given, to find the number of terms;

Divide the difference of the extremes by the common difference, and the quotient, increased by 1, will be the answer.

17. If the extremes be 3 and 45, and the common difference 6, what is the number of terms? A. 8.

18. A man, being asked how many children he had, replied, that the youngest was 4 years old, and the eldest 32, the increase of the family having been I in every 4 years; many

how

had he? A. 8.

GEOMETRICAL PROGRESSION.

¶ LXXXIX. Any rank or series of numbers, increasing by a constant multiplier, or decreasing by a constant divisor, is called Geometrical Progres

sion.

Thus, 3, 9, 27, 81, &c., is an increasing geometrical series;

And 81, 27, 9, 3, &c., is a decreasing geometrical series.

There are five terms in Geometrical Progression; and, like Arithmetica. Pro FECssion any three of them being given, the other two may be found, viz. 1. The first term.

2. The last term.

3. The number of terms.

4 The sum of all the terms.

. The ratio; that is, the multiplier or divisor, by which

1. A man purchased a flock of sheep, consisting of 9; and, by agreement was to pay what the last sheep came to, at the rate of $4 for the first sheep, $12 for the second, $36 for the third, and so on, trebling the price to the last what did the flock cost him?

We may perform this example by multiplication; thus, 4X3X 3X 3X3 X3X3X3X3: $26244, Ans. But this process, you must be sensible, would be, in many cases, a "ery tedious one; let us see if we cannot abridge it, thoroby making it easier.

In the above process we discover that 4 is multiplied by 3 eight times, one time iess than the number of terms; consequently, the 8th power of the ratio 3, expressed thus, 38, multiplied by the first term, 4, wil. produce the last term But, instead of raising 3 to the 8th power in this manner, we need only raise t to the 4th power, then multiply this 4th power into itself; for, in this way, wo dio, in fact, use the 3 eight times, raising the 3 to the same power as before; thus, 3 81; then, 81 X 81 6561; this, multiplied by 4, the first term, gives $26244, the same result as before. A $2624 1.

Hence, when the first term, ratio, and number of terms, are given, to find the last term;—

I. Write down some of the leading powers of the ratio, with the numbers 1, 2, 3, &c. over them, being their several indices. II. Add together the most convenient indices to make an index less by 1 thun the number of terms sought.

III. Multiply together the powers, or numbers standing under those indices; and their product, multiplied by the first term, will be the term sought.

2. If the first term of a geometrical series be 4, and the ratio 3, what is tl.o 11th term?

1, 2, 3, 4, 5, indices. Note. The pupil will notice that the series 3, 9, 27, 81, 213, powers. does not commence with the first term, but with the ratio.

The indices 5+3+2=10, and the powers under each, 243 X 27 X 9: 590-49; which, multiplied by the first term, 4, makes 236196, the 11th terin vequired. A. 236196.

3. The first term of a series, having 10 terms, is 4, and the ratio 3; what is the last term? A, 78732.

4. A sum of money is to be divided among 10 persons; the first to have $10, the second $30, and so on, in threefold proportion; what will the last have? A. $196830.

5. A boy purchased 18 oranges, on condition that ne should price of the last, reckoning 1 cent for the first, 4 cents for the second, 16 cents pay only the for the third, and in that proportion for the whole · how much did he pay for them? A $171798691,84.

6 What is the last term of a series having 18 terms, the first of which is 3, and the ratio 3? A. 387420489.

7. A butcher meets a drover, who has 24 oxen. The butcher inquires the price of them, and is answered, $60 per head; he immediately offers the drover $50 per head, and would take all. The drover says he will not take that; but, if he will give him what the last ox would come to, at 2 cents for the first, 4 cents for the second, and so on, doubling the price to the last, he might have the whole. What will the oxen amount to at that rate?

A. $167772,16

8. A man was to travel to a certain place in 4 days, and to travel at whatever rate he pleased; the first day he went 2 miles, the second 6 miles, and so on to the last, in a threefold ratio; how far did ho travel the last day, and

how far in all?

: In this example, we may find the last term as before, or find it by adding each day's travel together, commencing with the first, and proceeding to the last, thus: 26185480 miles, the whole distance travelled, and the last day's journey is 54 miles. But this mode of operation, in a long series, you must be sensible, would be very troublesome. Let us examine the na ture of the series, and try to invent some shorter method of arriving at the tamo result.

By examining the series 2, 6, 18, 54, we perceive that the last term, 154,) ess 2, (the first term,) == 52, is 2 times as large as the sum of the remaining erms; for 2+6+18=26; that is, 54-252226; hence, i we produce another term, that is, multiply 54, the last term, by the ratio 3 making 162, wo shall find the same true of this also; for 162-2, (the first term,) 160280, which we at first found to be the sum of the four emaining terms, thus: 2+6+18+5480. In both of theso operations it is curious to observe, that our divisor, (2,) cach time, is i less than the ra Lio, (3.)

Hence, when the extremes and ratio are given, to find the sum of the series, we have the following easy

RULE.

Multiply the last term by the ratio, from the product subtract the first term, and divide the remainder by the ratio, less 1; the quotient will be the sum of the series required.

9 If the extreines be 5 and 6-100, and the ratio 6, what is the whole amount of the series?

10. A sum of money is to be divided among 10 persons in such a manner that the first may have $10, the second $30, and so on, in threefold proportion; what will the last have, and what will the whole have?

The pupil will recollect how he found the last torm of the series by a foregoing rule; and, in all cases in which he is required to find the sum of the series, when the last term is not given, he must first find it by that rule, and then work for the sum of the series, by the present rule.

A. The last, $196830; and the whole, $2952-10

11. A hosier sold 14 pair of stockings, the first at 4 cents, the second at 12 cents, and so on in geometrical progression; what did the last pair bring him, and what did the whole bring him? A. Last, $63772,92; whole, $95659,36. 12. A man bought a horse, and, by agreement, was to give a cent for the first nail, three for the second, &c.; there were four shoes, and in each shoe ight nails; what did the horse come to at that rate?

A. $9265100944259,20

13. At the marriage of a lady, one of the guests made her a present of a ha f-eagle, saying, that he would double it on the first day of each succeeding mouth throughout the year, which he said would amount to something like $100; now, how much did his estimate differ from the true amount?

A. $20375. 14 If our pious ancestors, who landed at Plymouth, A. D. 1620, being 101 in number, had increased so as to double their number in every 20 years, how great would havɔ been their population at the end of the year 1840?

A. 206717