(h.) A FRUSTUм of a cone or pyramid is a part cut off by a plane parallel to the plane of its base.

(i.) Similar solids have the same shape, i. e., the angles of one of them equal the corresponding angles of the other, and the sides about the equal angles are proportional.

All spheres are similar. Two cones, or two cylinders, are similar when their altitudes are to each other as the radii or diameters of their

(j) Figure 1 represents a sphere; figure 2 a prism; figure 3 a cylinder; figure 4 a pyramid; figure 5 a cone; figure 6 a frustum of a

cone.

(k.) The SURFACE of a sphere equals the square of its diameter multiplied by 3.1416.*

(1.) The surfaces of spheres are to each other as the squares of their radii or diameters.

(m.) The SOLIDITY, or SOLID CONTENTS, of a sphere equals the product of the surface multiplied by of the radius, or by of the diameter, or it equals of the cube of the diameter multiplied by 3.1416.*

(n.) The solidities of spheres are to each other as the cubes of their radii or diameters.

*See foot note, page 343.

(o.) The solidities of similar solids are to each other as the cubes of their like dimensions.

(p.) The solidity of a prism equals the area of its base multiplied by its altitude.

(9.) The solidity of a cylinder is equal to the area of its base multiplied by its altitude.

(r) The convex surface of a cylinder is equal to the circumference of its base multiplied by its altitude.

(s.) The solidity of a cone or of a pyramid equals the area of its base multiplied by of its altitude.

(t.) The solidity of a frustum of a cone, or of a pyramid, equals ✯ of the product of its altitude multiplied by the sum of its upper base, plus its lower base, plus the mean proportional between the two bases.

NOTE. The mean proportional of two numbers is the square root of their product. Thus, the mean proportional of 4 and 9 = √4 X 9

= 6.

=

231. Problems.

1. What is the solidity of a sphere the diameter of which is 3 feet?

2. What is the surface of a sphere the radius of which is 1 foot?

3. What is the diameter of a sphere of which the solidity is 10 feet?

4. What is the circumference of a sphere the solidity of which is 12 feet?

5. What is the diameter of a sphere of which the surface is 6 feet?

6. What is the solidity of a prism of which the altitude is 9 feet, and the base contains 10 square feet?

7. What is the solidity of a cylinder of which the altitude is 6 feet and the radius of the base 2 feet?

8. What is the convex surface of a cylinder of which the diameter of the base is 5 feet and the altitude 4 feet?

9. What is the solidity of a cone of which the altitude is 9 feet and the circumference of the base 10 feet?

10. What must be the diameter of a sphere which contains 8 times as many cubic feet as one 3 feet ir diameter ?

SECTION XVIII.

PROGRESSIONS.

232. Arithmetical Progression.

(a.) A SERIES OF NUMBERS IN ARITHMETICAL progresSION, or an ARITHMETICAL SERIES, is a series of numbers each of which differs from the preceding by the same number. (b.) Such a series would be obtained by continually adding the same number to, or subtracting it from, any given number.

(c.) If the series is formed by addition, it is called an INCREASING SERIES; if by subtraction, it is called a DE

CREASING SERIES.

(d.) The numbers composing a series are called the TERMS of the series.

(e.) The difference between the consecutive terms of any series is called the COMMON DIFFERENCE, and is always the number by the addition or subtraction of which the series is formed.

(f) Since the terms of a series are formed by continual additions or subtractions of the same number, it follows that the second term of any series equals the first, plus or minus the common difference; that the third equals the first, plus or minus twice the common difference; that the fourth term equals the first, plus or minus three times the common difference; &c.

(g.) IIence, any term of an arithmetical series is equal to the first term, plus or minus the common difference taken one less times than there are terms in the series ending with the required term.

(h.) Moreover, if the first term of an increasing arithmetical series be subtracted from the last, or if the last term of a decreasing series be subtracted from the first, the remainder will be the product of the common difference multiplied by one less than the number of terms.

233. Problems.

1. What is the 10th term of the increasing series of which the first term is 3 and the common difference 8?

2. What is the 25th term of the decreasing series of which the first term is 85 and the common difference 2?

3. What is the common difference of the series of which the 1st term is 7 and the 13th term 43?

4. How many terms are there in the series of which the 1st term is 8, the last term 85, and the common difference 7? 5. What is the common difference of the series of which 596 is the 1st term and 491 the 22d?

6. How many terms are there in the series of which 12 is the first term, 4 the last, and the common difference?

234. To find the Sum of a Series.

(a.) If we should invert any series, we should have a new one, which would differ from the former only in the order of its terms, the one being an increasing while the other is a decreasing series. The first term of one series would equal the last of the other, and each term of one series would be as much greater than its preceding term as each of the other is less than its preceding term. Hence, if we should write the two series under each other, and add together the corresponding terms in the order in which they stand, the successive sums would equal each other, and each would equal the sum of the first and last terms of the original series.

(b.) Moreover, there would be as many such sums as there are terms in the series. Hence, the sum of the two series, or (since they are equal) twice the sum of either of them, is equal to the product obtained by multiplying the first and last terms by the number of terms.

Thus, by inverting the series 3, 7, 9, &c., to 35, we have,

3 7 11 15 19 23 27 31 35 = given series.

-

same series inverted.

the sums of the succes

(c.) Adding these last results together would give the sum of the two series, or twice the sum of either, which would manifestly be equal to 9 times 38, or 9 times the sum of the first and last terms. Dividing this by 2 would give the sum of one of the series.

(d.) Hence, the sum of a series in arithmetical progression equals half the product obtained by multiplying the sum of the first and last terms by the number of terms.

235. Problems.

1. What is the sum of the series of which 9 is the 1st terin and 94 the 20th?

(9 +94) × 20

Answer.

= 103 X 10 = 1030.

2

2. What is the sum of the series of which 427 is the 1st term and 187 the 81st?

3. What is the sum of the series of which 4 is the 1st term and 9 is the 6th? What is the common difference?

4. What is the sum of the series of which the 1st term is 7, the common difference 9, and the number of terms 15?

5. How many terms are there in a series of which the sum is 648, the 1st term 3, and the last term 78? What is the common difference?

6. What is the 1st term and common difference of a series of which the last term is 164, the number of terms 12, and the sum 2100?

7. Form the series of which the sum is 153, the 1st term 1, and the last term 17?

236. Geometrical Progression.

(a.) A series of numbers in geometrical progression, or a geometrical series, is a series of numbers each of which bears the same ratio to the one which follows it.

(b.) Such a series would be obtained by continually multiplying or dividing by the same number.

Thus, beginning with 2 and multiplying by 3, we should have 2, 6, 18, 54, 162, 486, &c.

By beginning with 3072 and multiplying by, we have 3072, 1536, 768, 384, 192, 96, 48, &c.

(c.) The numbers comprising the series are called the

TERMS OF THE SERIES.