14 inches 1 feet of a foot.

7×3×0-7854=1.069 square feet for area of end. 1.069 × 14 14.966 cubic feet.

3. How many cubic inches in a round bar of iron, 20 leet long and 4 of an inch in diameter ?

Ans. 106 029 cubic inches.

PROBLEM VIII.-To find the volume of a pyramid, or of

cone.

RULE.

Multiply the area of the base by one-third the altitude. (Geometry, B. VII., Prop. XVII.; and B. VIII. Prop. V.)

EXAMPLES.

1. The Egyptian pyramid, Cheops, covers a square of 763 feet on a side, and is 480 feet perpendicular height. How many cubic feet does it contain?

Ans. 93244729 cubic feet. 2. Suppose the mast of a ship to be a regular cone 87 feet long, and 2 feet in diameter at its base, how many cubic feet will it contain? Ans. 91-1064 cubic feet.

PROBLEM IX-T find the surface of a sphere, when its diameter is given.

RULE.

Multiply the square of the diameter by 3-1416. (Geometry

B VIII., Prop. XIII. Schol.)

EXAMPLES.

1. How many square miles on the surface of the earth, on the supposition that it is an exact sphere of 8000 miles in diameter ?

Ans. 8000 x 8000 x 3.1416-201062400 square miles.

In order to obtain a value true to a unit, we must use, for our multiplier, 3.14159265, instead of 3-1416.

Using this more accurate value, we find the

Ans. 201061930 square miles, nearly. 2. How many superficial inches has a ball 6 inches in Ans. 113-0976 square inches.

diameter ?

PROBLEM X.-To find the volume of a sphere, when its

diameter is given.

RULE.

Multiply the cube of the diameter by 0·5236, which is of 3.1416. (Geometry, B. VIII., Prop. XIII. Schol.)

1. How many cubic inches in a ball 6 inches in diameter? Ans. 6×6×6×0·5236=113·0976 cubic inches.

NOTE.-Comparing this Example with Example 2, under last Problem, we see that the number of superficial inches and cubie inches are equal in a sphere of 6 inches in diameter.

2. How many cubic inches in a ball of the celebrated Stockton gun, the diameter of which is 12 inches?

Ans. 904-7808 cubic inches.

The following table of multipliers will be found very convenient for solving nearly all problems which can arise in mensuration of circles and spheres.

TABLE OF MULTIPLIERS.

1. Radius of a circle X6-28318531-Circumference.
2. Square of the radius of a circle X3-14159265—Area.
3. Diameter of a circle X3-14159265-Circumference.
4. Square of the diameter of a circle X0-78539816=Area.
5. Circumference of a circleX0-15915494-Radius.
6 Circumference of a circleX0-31830989-Diameter.
7 Square root of area of a circle X0.56418958-Radius.
8. Square root of area of a circle X1·12837917-Diameter.

9. Radius of circle X1·73205081 Side of inscribed equilateral triangle
10. Side of inscribed equilateral triangle×0.57735027=Radius of circie.
11. Radius of a circle X1-41421356-Side of inscribed square.

12. Side of inscribed square X0-70710678=Radius.
13. Square of radius of a sphereX 12-56637061=Surface
14. Cube of radius of a sphereX4.18879020=Volume.
15. Square of diameter of a sphereX3.14159265 Surface.
16. Cube of diameter of a sphereX0·52359878=Volume.
17. Square of circumference of a sphereX0-31830989 Surface.
18. Cube of circumference of a sphereX001688686 Volume.
19. Square root of surface of a sphereX0-28209479 Radius.
20. Square root of surface of a sphereX0-56418958 Diameter.
21. Square root of surface of a sphereX1-77245385—Circumference.
22. Cube root of volume of a sphereX0-62035049=Radius.
23. Cube root of volume of a sphereX1·24070098=Diameter.
24. Cube root of volume of a sphereX3-89777707 Circumference.
25. Radius of a sphereX1·15470054-Side of inscribed cube.
26. Side of inscribed cubeX0-86602540 Radius.

PROBLEM XI. To find the volume of a frustum of a pyra mid, or of a cone.

RULE.

Find a mean proportional between the area of the two bases, to which add the sum of the bases, and multiply the result by one-third the altitude of the frustum.

EXAMPLES.

1. Suppose a cistern in the form of a frustum of

cone, to be 9 feet deep, having for diameters 8 feet and 10

feet.

How many cubic feet will it contain?

10a ×0 7854=100×0-7854-area of one base

And 1960-7854x of 9=461-8152 cubic feet, for its volume.

2. Suppose a measure to be in the form of a frustum of a regular cone. If its top diameter is 6 inches, and the bottom diameter 9 inches, and it is 12 inches deep, how many cubic inches will it contain? and how many beer gallons of 282 cubic inches each ?

PROBLEM XII.—To find the area of an ellipse.

NOTE-A line drawn through the centre of an ellipse is called its diameter. The longest diameter is called the transverse diameter; the shortest is called the conjugate diameter. Thus AB is the transverse diameter, and CD is the conjugate diameter.

D

The area of an ellipse may be found by this

RULE.

Multiply the product of the transverse and conjugate

diameters by 0.7854.

.

1. How many square feet in the surface of an elliptical pond, whose transversé diameter is 100 feet, and conju gate diameter 60 feet?

Ans. 100 × 60 × 0.7854-4712-4 square feet.

2. How many square inches is an elliptical table whose transverse diameter is 5 feet 3 inches, and conjugate diam. eter 3 feet 6 inches ? And how many square feet? 2078 1684 square inches. 14.4317 square feet.

Ans.

{

NOTE 1.-If an ellipse be inscribed in a rectangle, its area will be to the area of the rectangle as 0-7854 is to 1.

NOTE 2.-We also infer that, if a circle be inscribed in an ellipse, and another circle be circumscribed about the same ellipse, the ellipse is a mean proportional between the areas of the two circles; that is, we' shall have, area of inscribed circle is to the area of ellipse, as area of ellipse is to the area of circumscribed circle.

PROMISCUOUS QUESTIONS.

144. 1. Suppose I purchase $1200 worth of goods, of which is on a credit of 3 months, on a credit of 6