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

C

ART. 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.

ART. 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 proportion is the required answer.

ART. 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 height contains 456 cubic feet, what are the contents of a similar cone, the altitude of which is 3 feet? Ans. 1539 cubic feet.

OPERATION.

23:33:456:1539.

2. If a cubic piece of metal, the side of which is 2 feet, is worth $6.25, what is another cubical piece of the same kind worth, one side of which is 12 feet? Ans. $1350.

3. If a ball, 4 inches in diameter, weighs 50lb., what is the weight of a ball 6 inches in diameter ? Ans. 168.7+lb.

QUESTIONS. What is the altitude of a cone? Art. 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? - Art. 301. What is the rule for finding the contents of a solid similar to a given solid? Art. 302. What is 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 height, 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 one foot in diameter, make one cord, how many cords are there in a similar tree, whose diameter is two 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.

§ XXXIX. ARITHMETICAL PROGRESSION.

ART. 303. WHEN a series of numbers increases or decreases by a constant difference, it is called Arithmetical Progression, or Progression by Difference. Thus,

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

The first is called an ascending series or progression. The second is called a descending series or progression. The numbers which form the series are called the terms of the progression. The first and last terms are called the extremes, and the other terms the means. The constant difference is called the common difference of the progression.

Any three of the five following things being given, the other two may be found:

QUESTIONS. Art. 303. What is arithmetical progression? What is an ascending series? What a descending series? What are the terms of a progression? What the extremes? What the means?

ART. 304.

To find the common difference, the first term, last

term, and number of terms, being given.

ILLUSTRATION. In the following series,

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 9, and their sum will be equal to the difference of the extremes; hence, if the difference of the extremes (29 227) be divided by the number of common differences, the quotient will be the common difference. Thus, 279 = 3, the common difference. Hence the following

RULE.

Divide the difference of the extremes by the number of term less one, and the quotient is 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,

QUESTIONS. Art. 304. What is the common difference? What five things are named, any three of which being given the other two can be found. What is the rule for finding the common difference, the first term, last term, and number of terms, being given?

and the last 17 miles, and increases each day's travel by an equal difference, what is the daily increase ? Ans. 8 miles. ART. 305. To find the sum of all the terms, the first term, last term, and number of terms, being given.

ILLUSTRATION.Let the two following series be arranged as

follows:

2, 5, 8, 11, 14, 17, 20, 20, 17, 14, 11, 8,

8, 5,

2,

= 77, sum of first series.
= 77, sum of inverted series.

22, 22, 22, 22, 22, 22, 22, 154, sum of both series.

=

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 × 7: 154; and, therefore, the sum of either series will be 1542=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.

RULE 2.

Or,

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

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?

(455) X9
2

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.

QUESTION. Art. 305. What is 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 travelled 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.

ART. 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 29-227. 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, 273: It has been shown (Art. 303) 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 following

=

= 9.

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

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 travelled 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.

ART. 307. To find the sum of the series, 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

QUESTION.Art. 306. What is the rule for finding the number of terms, the extremes and common difference being given?