of the whole surface of the cylinder; also, the solidity of the sphere is two thirds of that of the circumscribed cylinder.

Let ABFI be a great circle of the sphere; DEGH the circumscribed square; then, if the semicircle ABF and the semi-square ADEF be revolved about the diameter AF, the semicircle will describe a sphere, and the semisquare a cylinder circumscribing the sphere.

D

B

A

H

G

F

The convex surface of the cylinder is equal to the circumference of its base multiplied by its altitude (Prop. I.). But the base of the cylinder is equal to the great circle of the sphere, its diameter E G being equal to the diameter BI, and the altitude DE is equal to the diameter AF; hence, the convex surface of the cylinder is equal to the circumference of the great circle multiplied by its diameter. This measure is the same as that of the surface of the sphere (Prop. VIII.); hence, the surface of the sphere is equal to the convex surface of the circumscribed cylinder.

But the surface of the sphere is equal to four great circles of the sphere (Prop. VIII. Cor. 1); hence, the convex surface of the cylinder is also equal to four great circles; and adding the two bases, each equal to a great circle, the whole surface of the circumscribed cylinder is equal to six great circles of the sphere; hence, the surface of the sphere isor of the whole surface of the circumscribed sphere.

In the next place, since the base of the circumscribed cylinder is equal to a great circle of the sphere, and its altitude to the diameter, the solidity of the cylinder is equal to a great circle multiplied by its diameter (Prop. II.). But the solidity of the sphere is equal to its sur

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.

TABLE OF LINEAR MEASURES.

610.

NOTE.

For other linear measures, see National Arithmetic, Art.

133, 134, 136.

611.

TABLE OF SURFACE MEASURES.

144 Square Inches make 1 Square Foot.

66

10 Square Chains 1 Acre.

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 Α 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

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. 64 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.

354 X 192 67,968 feet, Ans.

=

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