5 The hypothenuse of a right-angled triangle is 40 feet; what must be the hypothenuse of a similar triangle that it may contain twice the area? Ans. 56.568+.

6. If a circular lot of land which is 10 rods in diameter contains 78.5398 square rods, what number of rods will a lot contain which is 5 rods in diameter? Ans. 19.63495.

7. The area of a triangle whose base is 24 feet is 120 feet; what is the area of a similar triangle whose base is 96 feet? Ans. 1920 feet.

8. The Winchester bushel is 18 inches in diameter and 8 inches deep; what must be the diameter of a circular measure 6 inches deep, that it may hold a bushel? Ans. 21.36 inches. 9. I have a circular flower-garden, the circumference of which is bordered with 75 yards in length of sodding; how many yards will be required to border a circular garden of the area?

Ans. 61.237+ yards. 10. Having a triangular board 7 feet long, what distance from the base end shall I cut it to divide it into two equal parts? Ans. 2.197-ft.

SIMILAR SOLIDS.

NOTE.Angular solids are similar when their angles are equal each to each, and arranged in the same way, and their corresponding edges proportional.

The following proposition may be easily proved by geometry:

472. The Solidities of Cubes, Spheres, and all Similar Solids are to each other as the cubes of their corresponding dimensions.

ILL. EX., I. How many globes of 6 inches diameter can be made from a globe of 48 inches diameter ?

By Proportion, 63 : 483 = 1 (globe) : 512 globes, Ans.

ILL. EX., II. If a conical stack of hay which contains of a ton is 6 feet high, what is the height of a similar stack which contains 3 tons?

By Proportion,

: 363: 1728, the cube of the height of the larger stack.

172812 ft., height of larger stack, Ans.

EXAMPLES.

1. If an ounce ball is § inch in diameter, how many ounce balls can be made from a globe of lead 6 inches in diameter?

Ans. 884 balls. 2. A pyramid which is 9 feet in height contains 48 cubic feet; what is the height of a similar pyramid that contains 100 cubic feet? Ans. 11.494

feet. 3. If a cube of granite whose edge is 2 feet weighs 1336.32 pounds, what will be the weight of a cube whose edge is 4 feet, 9 inches? Ans. 17901.99. 4. If an egg of 24 inches in circumference weigh 1 ounce, what would another of the same form and consistency weigh whose circumference is 6 inches?

5. What must be the height of a cone to contain 125 times as many solid inches as a similar cone 3 inches in height?

Ans. 15 inches. 6. If a bushel measure is 18 inches in diameter and 8 inches deep, what must be the diameter and depth of a half-bushel measure similar in form?

Ans. Diam. 14.683+ in.; depth 6.349+ in. 7. If an elephant's tusk 94 feet long and 8 inches in diameter at base weigh 214 pounds, what would be the dimensions of a similar tusk weighing 75 pounds?

8. Estimating the mean diameter of the earth at 7912 miles, and that of the moon at 2160 miles, how many bodies of the size of the moon could be made from the bulk of the earth?

9. If the bulk of Saturn be 1000 times as great as that of the earth, what is the diameter of Saturn?

10. At what distance from the top, parallel with the base, must conical sugar-loaf 12 inches high be cut that it may be divided into two equal parts?

11. Mr. Root has three stacks of hay of similar shape, the

diameters of their bases being, respectively, 10, 12, and 14 feet; if the one whose diameter is 10 feet contains 2 tons, what will each of the others contain?

For Dictation Exercises, see Key.

473. QUESTIONS FOR REVIEW.

What is Involution? a power? an exponent? What is the first power? second power? third power? fourth power?

What are the second and third powers generally called?

Rule for Involution? How may a mixed number be raised to a given power? Repeat the squares of the integers from 1 to 25;* the cubes of the integers from 1 to 10.*

How does Evolution differ from Involution?

What is a root? What is the square root of a number? the cube root? How is the square root indicated? the cube root? How otherwise may the root of a number be indicated?

If a power contain one or two figures only, of how many figures will its square root consist? If a power contain three or four figures? If five or six ?

What three terms does every square number contain whose root consists of tens and units?

Give the rule for extracting the square root. Of how many figures may the left-hand period in whole numbers consist? of how many must every period, except this, consist? of how many, every period in decimals?

How do you proceed when a zero occurs in the root? how when a root figure proves too large? how when there is a remainder? How do you extract the square root of a common fraction whose terms are square numbers? whose terms are not squares? How extract the root of a mixed number? Explain the extraction of the square root by an example. Illustrate by diagrams.

What is an angle? a right angle? a triangle? a right-angled triangle? its hypothenuse? its perpendicular? its base?

To what is the square on the hypothenuse of a right-angled triangle equal? Rule to find the hypothenuse; to find base or perpendicular. If a cube number contain one, two, or three figures only, of how many figures will its cube root consist? if it contain four, five, or six figures only?

To what four terms is every cube number equal whose root consists of tens and units?

* At the option of the teacher.

Give the rule for extracting the cube root. Of how many figures may the left-hand period consist? of how many must every other period consist in whole numbers? in decimals?

How do you proceed when a zero occurs in the root? how when a root figure proves too large? how when there is a remainder? How do you extract the cube root of a common fraction when the terms are cubes? how when the terms are not? How extract the cube root of a mixed number?

Explain the extraction of the cube root by an example. Illustrate by blocks.

What is Mensuration ? Name and describe the different kinds of triangles. Draw a right-angled triangle; an obtuse-angled triangle; an equilateral triangle; an isosceles triangle; a scalene triangle. Name and describe the different kinds of quadrilaterals. Draw a square; a rectangle; a rhombus; a trapezoid; a trapezium; a circle; a polygon of 5 sides with 2 diagonals.

How do you find the area of a square? a rectangle? any parallelogram? a triangle? a trapezoid? a trapezium? any polygon?

How do you find the circumference of a circle when the diameter is given? when the radius is given? How do you find the area of a circle when the diameter is given? when the radius is given? How do you find the diameter when the circumference is given? How find diameter when the area is given? How do you find the radius when the area is given?

cylinder; pyramid; cone; Draw or mention someWhat is the height of any

Define a cube; parallelopiped; prism; frustum of a pyramid or cone; a sphere. thing in the form of each of these solids. solid? the slant height of a pyramid or cone? the slant height of a frustum of a pyramid or cone?

How do you obtain the solid contents of a cube? a parallelopiped? a prism or cylinder? a pyramid or cone? a frustum of a pyramid or cone? How do you find the convex surface of each of these solids?

When the diameters and altitude are given, how do you find the solid contents of a cylinder? of a cone? of a frustum of a cone? of a sphere? How do you find the convex surface of a cylinder? of a cone? the frustum of a cone? a sphere ?

What proportion exists between the areas of squares? of circles? of all similar triangles and polygons? When are angular figures similar? What proportion exists between spheres ? between all similar solids? When are angular solids similar?

474. GENERAL REVIEW, No. 8.

1. Supply the 2d term in the proportion 34: ?

8:25 X 6. 2. What is the mean proportional between .8 and .72?

3. Divide $1900 between two men, in the proportion of 3 to 5. 4. Divide $45 between three boys, so that one shall have as much as the other two, whose shares are as 2 to 7.

5. If 15 gallons of oil cost 7 £, 10 s., what will 25 gallons cost? 6. How many pounds can 5 horses draw, if 6 horses can draw as much as 10 oxen, and 2 oxen can draw 2400 pounds?

7. Smith and Lee formed a partnership. Smith put in $1000 for 6 months, and Lee $600 and his services for 8 months, his services being equal to $100 a month. They gained $1506; what was each one's share?

8. What is the 5th power of 23? the cube of 96?

9. What is the largest number of men in a regiment of 1000 that can be arranged in a square; and how many men will remain? How many men will there be on each side of the square?

+

10. How many feet of fencing around a square farm containing 15 acres?

11. A ladder 279 feet long reaches a window 25 feet from the ground; how far does the foot of the ladder stand from the house?

12. Required the diameter of a circular grass plat which contains 314 square feet.

13. How many rods of fencing on both sides of a road which surrounds a circular park containing of a square mile, the road being 3 rods wide?

14. What must be the depth of a pail, that is 10 inches across, to contain 5 gallons (the sides being upright)?

15. How many feet of canvas are required to construct a conical tent 14 feet across the bottom and 94 feet from the highest point to the ground?

16* How many gallons will a circular vat contain, which measures across the top 8 ft., across the bottom 7 ft., the sides sloping uniformly and measuring on the slope 64 ft. in depth?

For changes, see Key.