The Altitude of a cone is its perpendicular hight, or a line drawn from the vertex perpendicular to the plane of the base; as B C.

300. SPHERES are to each other as the cubes of their diameters, or of their circumferences.

SIMILAR CONES are to each other as the cubes of their altitudes, or the diameters of their bases.

All SIMILAR SOLIDS are to each other as the cubes of their homologous or corresponding sides, or of their diameters.

301. To find the contents of any solid which is similar to a given solid.

RULE.

State the question as in Proportion, and cube the given sides, diameters, altitudes, or circumferences, and the fourth term of the propor tion is the required answer.

302. To find the side, diameter, circumference, or altitude, of any solid, which is similar to a given solid.

RULE.

State the question as in Proportion, and cube the given sides, diameters, circumferences, or altitudes, and the cube root of the fourth term of the proportion is the required answer.

EXAMPLES FOR PRACTICE.

1. If a cone 2 feet in hight contains 456 cubic feet, what are the contents of a similar cone, the altitude of which is 3 feet? Ans. 1539 cubic feet.

OPERATION.

23:33:45 6:1539

2. If a cubic piece of metal, the side of which is 2 feet, is worth $6.25, what is another cubical piece worth, one side of which is 12 feet?

3. If a ball, 4 inches in diameter, weighs weight of a ball 6 inches in diameter ?

of the same kind

Ans. $1350. 50lb., what is the Ans. 168.7+lb.

-301. What is the rule

299. What is the altitude of a cone? 300. What proportion do spheres have to each other? What proportion do cones have to each other? What proportion do all similar solids have to each other? for finding the contents of a solid similar to a given solid?-302. The rule for finding the side, diameter, &c., of a solid similar to a given solid?

4. If a sugar-loaf, which is 12 inches in hight, weighs 16lb., how many inches may be broken from the base, that the residue may weigh 81b.? Ans. 2.5 in.

5. If an ox, that weighs 800lb., girts 6 feet, what is the weight of an ox that girts 7 feet? Ans. 1270.3lb.

6. If a tree, that is 1 foot in diameter, make 1 cord, how many cords are there in a similar tree, whose diameter is 2 feet? Ans. 8 cords.

7. If a bell, 30 inches high, weighs 1000lb., what is the weight of a bell 40 inches high? Ans. 2370.3lb. 8. If an apple, 6 inches in circumference, weighs 16 ounces, what is the weight of an apple 12 inches in circumference? Ans. 128 ounces.

9. A and B own a stack of hay in a conical form. It is 15 feet high, and A owns of the stack; it is required to know how many feet he must take from the top of it for his share. Ans. 13.1+ feet.

303.

ARITHMETICAL PROGRESSION.

Arithmetical Progression is a series of numbers that in

creases or decreases by a constant difference.

The Terms of the series are numbers of which it is formed.

The Extremes are the first and last terms.

The Means are the terms between the extremes.

The Common Difference is the constant difference between the terms.

The series is ascending when each term after the first exceeds that before it, and descending when each term after the first is less than that before it.

Thus, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, is an ascending series, and 29, 26, 23, 20, 17, 14, 11, 8, 5, 2, is a descending series.

303. What is arithmetical progression? gression? The extremes? The means? an ascending series? A descending series?

What are the terms of a pro-
Common difference? What is

In Arithmetical Progression, the first term, the last term, the number of terms, the common difference, and the sum of the terins, are so related to each other, that any three of these being given, the two others may be readily determined.

304. To find the COMMON DIFFERENCE, the first term, last term, and number of terms being given.

2, 5, 8, 11, 14, 17, 20, 23, 26, 29,

2 and 29 are the extremes, 3 the common difference, 10 the number of terms, and the sum of the series 155.

It is evident that the number of common differences in any series must be 1 less than the number of terms. Therefore, since the number of terms in this series is 10, the number of common differences will be 10-1 = 9, and their sum will be equal to the difference of the extremes; hence, if the difference of the extremes (29 — 2 = 27) be divided by the number of common differences, 9, the quotient, 3, will be the common difference. Hence the

RULE. — Divide the difference of the extremes by the number of terms less one, and the quotient will be the common difference.

EXAMPLES FOR PRACTICE.

1. The extremes of a series are 3 and 35, and the number of terms is 9; what is the common difference? Ans. 4.

2. If the first term is 7, the last term 55, and the number of terms 17, required the common difference.

Ans. 3.

3. If the first term is 4, the last term 14, and the number of terms 15, what is the common difference? Ans..

4. If a man travels 10 days, and the first day goes 9 miles, and the last 17 miles, and increases each day's travel by an equal difference, what is the daily increase? Ans. & miles.

303. What five things are named, any three of which being given the other two can be found? 304. The rule for finding the common difference, the

first term, last term, and number of terms being given?

305. To find the SUM OF ALL THE TERMS, the first term, last term, and number of terms being given.

Let the two following series be arranged as

ILLUSTRATION.

From the arrangement of the above series, we see that, by adding the two as they stand, we have the same number for the sum of the successive terms, and that the sum of both series is double the sum of either series.

=

It is evident that, if 22 in the above series be multiplied by 7, the number of terms, the product will be the sum of both series; thus, 22 X 7 154; and, therefore, the sum of either series will be 154 ÷ 2 77. But 22 is the sum of the extremes in each series; thus, 20 +2 22. Therefore, if the sum of the extremes be multiplied by the number of terms, the product will be double the sum of either series. Hence,

=

=

RULE 1. - Multiply the sum of the extremes by the number of terms and half the product will be the sum of the series. Or,

RULE 2.

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

EXAMPLES FOR PRACTICE.

1. If the extremes of a series are 5 and 45, and the number of terms 9, what is the sum of the series?

(45+5) X9
2

OPERATION.

225, sum of the series.

Ans. 225.

2. John Oaks engaged to labor for me 12 months. For the first month I was to pay him $7, and for the last month $51. In each successive month he was to have an equal addition to his wages; what sum did he receive for his year's labor?

Ans. $348.

305. The rule for finding the sum of all the terms, the first term, last term, and number of terms being given?

3. I have purchased from W. Hall's nursery 100 fruit-trees of various kinds, to be set around a circular lot of land at the distance of one rod from each other. Having deposited them on one side of the lot, how far shall I have traveled when I have set out my last tree, provided I take only one tree at a time, and travel on the same line each way? Ans. 9801 rods.

306. To find the NUMBER OF TERMS, the extremes and common difference being given.

ILLUSTRATION. Let the extremes of a series be 2 and 29, and the common difference 3. The difference of the extremes will be 29 - 2 = 27. Now, it is evident that, if the difference of the extremes be divided by the common difference, the quotient will be the number of common differences; thus, 27 ÷ 3 = 9. It has been shown (Art. 304) that the number of terms is 1 more than the number of differences; therefore 9+ 1 10 is the number of terms in this series. Hence the

=

RULE. Divide the difference of the extremes by the common differ ence, and the quotient, increased by 1, will be the number of terms.

EXAMPLES FOR PRACTICE.

1. If the extremes of a series are 4 and 44, and the common difference 5, what is the number of terms?

Ans. 9.

2. A man going a journey traveled the first day 8 miles, and the last day 47 miles, and each day increased his journey by 3 miles. How many days did he travel? Ans. 14 days.

307. To find the SUM OF THE TERMS, the extremes and common difference being given.

ILLUSTRATION. - Let the extremes be 2 and 29, and the common difference 3. The difference of the extremes will be 29 2 = 27; and it has been shown (Art. 306) that if the difference of the extremes be divided by the common difference, the

306. The rule for finding the number of terms, the extremes and common difference being given?