face, or four great circles, multiplied by one third of its radius (Prop. IX.), which is the same as one great circle multiplied by of the radius, or by of the diameter; hence, the solidity of the sphere is equal to of that of the circumscribed cylinder.

605. Cor. 1. Hence the sphere is to the circumscribed cylinder as 2 to 3; and their solidities are to each other as their surfaces.

606. Cor. 2. Since a cone is one third of a cylinder of the same base and altitude (Prop. V. Cor. 1), if a cone has the diameter of its base and its altitude each equal to the diameter of a given sphere, the solidities of the cone and sphere are to each other as 1 to 2; and the solidities of the cone, sphere, and circumscribing cylinder are to each other, respectively, as 1, 2, and 3.

BOOK XI.

APPLICATIONS OF GEOMETRY TO THE MENSURATION OF PLANE FIGURES.

DEFINITIONS.

607. MENSURATION OF PLANE FIGURES is the process of determining the areas of plane surfaces.

608. The AREA of a figure, or its quantity of surface, is determined by the number of times the given surface contains some other area, assumed as the unit of measure.

609. The MEASURING UNIT assumed for a given surface is called the superficial unit, and is usually a square, taking its name from the linear unit forming its side; as a square whose side is 1 inch, 1 foot, 1 yard, &c.

Some superficial units, however, have no corresponding linear unit; as the rood, acre, &c.

610. TABLE OF LINEAR MEASURES.

Inches make 1 Foot.

12

NOTE. For other linear measures, see National Arithmetic, Art.

133, 134, 136.

611.

TABLE OF SURFACE MEASURES.

144 Square Inches make 1 Square Foot.

612. Since an acre is equal to 10 chains, or 100,000 links, square chains may be readily reduced to acres by pointing off one decimal place from the right, and square links by pointing off five decimal places from the right.

PROBLEM I.

613. To find the area of a PARALLELOGRAM.

Multiply the base by the altitude, and the product will be the area (Prop. V. Bk. IV.).

EXAMPLES.

1. What is the area of a square, A B C D, whose side is 25 feet?

25 X 25625 feet, Ans.

2. What is the area of a square field whose Ꭺ . side is 35.25 chains? Ans. 124 A. 1R. 1P.

B

3. How many square feet of boards are required to lay

a floor 21 ft. 6 in. square?

4. Required the area of a square farm, whose side is 3,525 links.

5. What is the area of the rectangle D

ABCD, whose length, A B, is 56 feet,

and whose width, AD, is 37 feet?

C

56 × 372,072 feet, Ans.

A

B

6. How many square feet in a plank, of a rectangular form, which is 18 feet long and 1 foot 6 inches wide? 7. How many acres in a rectangular garden, whose / sides are 326 and 153 feet? Ans. 1 A. 23 P. 61 yd.

8. A rectangular court 68 ft. 3 in. long, by 56 ft. 8 in. broad, is to be paved with stones of a rectangular form, each 2 ft. 3 in. by 10 in.; how many stones will be required? Ans. 2,062 stones.

9. Required the area of the rhomboid A B C D, of which the side A B is 354 feet, and the perpendicular distance, E F, between A B and the opposite side CD, is 192 feet.

D E

C

A

F B

10. How many square feet in a flower-plat, in the form of a rhombus, whose side is 12 feet, and the perpendicular distance between two opposite sides of which is 8 feet?

11. How many acres in a rhomboidal field, of which the sides are 1,234 and 762 links, and the perpendicular distance between the longer sides of which is 658 links? Ans. 8 A. 19 P. 4 yd. 6 ft.

PROBLEM II.

614. The area of a SQUARE being given, to find the side. Extract the square root of the area.

Scholium. This and the two following problems are the converse of Prob. I.

EXAMPLES.

1. What is the side of a square containing 625 square feet?

✔625 25 feet, the side required.

=

2. The area of a square farm is 124 A. 1 R. 1 P.; how many links in length is its side?

3. A certain corn-field in the form of a square contains

15 A. 2 R. 20 P. If the corn is planted on the margin, 4 hills to a rod in length, how many hills are there on the margin of the field? Ans. 800 hills.

PROBLEM III.

615. The area of a RECTANGLE and either of its sides being given, to find the other side.

Divide the area by the given side, and the quotient will be the other side.

EXAMPLES.

1. The area of a rectangle is 2,072 feet, and the length of one of the sides is 56 feet; what is the length of the other side?

2072 56 37 feet, the side required.
÷ =

2. How long must a rectangular board be, which is 15 inches in width, to contain 11 square feet?

3. A rectangular piece of land containing 6 acres is 120 rods long; what is its width? Ans. 8 rods. 4. The area of a rectangular farm is 266 A. 3 R. 8P., and the breadth 46 chains; what is the length?

PROBLEM IV.

Ans. 58 chains.

616. The area of a RHOMBOID or RHOMBUS and the length of the base being given, to find the altitude; or the area and the altitude being given, to find the base.

Divide the area by the length of the base, and the quotient will be the altitude; or divide the area by the altitude, and the quotient will be the length of the base.

EXAMPLES.

1. The area of a rhomboid is 67,968 square feet, and the length of the side taken as its base 354 feet; what is the altitude?

67,968354192 feet, the altitude required.

2. The area of a piece of land in the form of a rhombus