12. What is the cube root of 2 to three places of decimals? Ans. 1.259+

13. What is the cube root of 517 to three places of decimals? Ans. 8.025+.

400. When the given number is a common fraction, or a mixed number, reduce it to its simplest form, and, if the numerator and denominator are both perfect cubes, extract the cube root of each separately; but, if not, reduce the fraction to an equivalent decimal, and extract the root of it.

1. A block of granite in the form of a cube contains 103823 cubic inches; what is the measure of one of its equal edges? Ans. 47 inches.

2. Required the depth of a cubic box that shall exactly hold a bushel. Ans. 12.9 inches.

3. There is a range of wood 213 feet long, 6 feet high, and 4 feet wide; how long a cubic pile will it make? Ans. 8 feet. 4. Find the area of a side of a cube containing 474552 liters. Ans. 60.84 square meters

5. There is a cistern of a cubical form, which contains 1331 cubic feet; what are the length, breadth, and depth of it? 6. What must be the depth of a cubical cistern that shall contain 576 gallons?

Ans. 4.25

feet.

How do we proceed when the given number is a common fraction or a mixed number?

MENSURATION.

401. A Point is that which has only position.

402. A Line is that which has only length.

A STRAIGHT LINE is one that has all its parts in the same direction.

A CURVED LINE is one that continually changes its direction.

STRAIGHT LINE.

CURVED LINE.

403. A Plane is a surface (Art. 199) in which any two points being taken, the straight line that joins them will lie wholly in the surface.

A CURVED SURFACE is one of which no part is plane.

404. Parallel Lines are such as, being

in the same plane, have the same direction

with each other.

PARALLEL LINES.

405. Two straight lines are said to be perpendicular to each other when their meeting forms equal adjacent angles (Art. 208).

406. A Right Angle is one formed by a straight line and a perpendicular to it.

TWO RIGHT ANGLES.

OBTUSE ANGLE. ACUTE ANGLE.

An Acute Angle is one which is less than a right angle.

What is a Point? A Line? A Straight Line? A Curved Line? A Plane? A Curved Surface? Parallel Lines? A Right Angle? An Acute Angle?

An Obtuse Angle is one which is greater than a right angle.

The sides of an angle are the lines forming it, and the vertex of an angle the point of their meeting.

407. A Plane Figure is a plane bounded by a line or lines.

The PERIMETER of a plane figure is its boundary.

The BASE of a figure is the line upon which it is supposed to stand.

The ALTITUDE of a figure is its perpendicular height.

The DIAGONAL of a figure is a straight line joining any two of its angles, which are not adjacent to each other.

The sides of a figure bounded by straight lines are the bounding lines.

408. A Polygon is a plane figure bounded by straight lines. A REGULAR POLYGON has equal sides and equal angles.

A Triangle is a polygon having three sides.

A Quadrilateral is a polygon having four sides.
A Pentagon is a polygon having five sides.
A Hexagon is a polygon having six sides.
A Heptagon is a polygon having seven sides.
An Octagon is a polygon having eight sides.

409. Mensuration treats of the measurement of lines, planes, and solids or volumes (Art. 199).

What is an Obtuse Angle? A Plane Figure? The Perimeter of a plane figure? The Base? The Altitude? The Diagonal? What is a Polygon? A Regular Polygon? Mensuration?

TRIANGLES.

410. A Triangle is a polygon having three sides, and therefore three angles.

An ACUTE-ANGLED TRIANGLE has three acute angles.

An OBTUSE-ANgled TriangLE has one obtuse angle.

ACUTE-ANGLED TRIANGLE.

OBTUSE-ANGLED TRIANGLE.

A RIGHT-ANgled TriangLE has one right angle.

The side opposite the right angle is called the HYPOTHENUSE, and the side perpendicular

to the base, the PERPENDICULAR.

RIGHT-ANGLED TRIANGLE.

411. By Geometry there may be readily demonstrated the following

PRINCIPLES.

1. The square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. Thus,

Let h denote the hypothenuse, b the base, and p the perpendicular of a right-angled triangle, and we have the formula,

h2 = b2 + p2,

which is illustrated by the diagram.

What is a Triangle? An Acute-Angled Triangle? An Obtuse-Angled Triangle? A Right-Angled Triangle? What are the sides called? What is the first Principle?

2. The hypothenuse of a right-angled triangle is equal to the square root of the sum of the squares of the other two sides; and

3. Either of the two shorter sides of a right-angled triangle is equal to the square root of the difference of the squares of the hypothenuse and the other side.

Exercises.

1. If the base of a right-angled triangle is 60 feet, and the perpendicular 45 feet, what is the hypothenuse?

OPERATION.

602+452 = 3600+ 2025 = 5625; 5625 = 75, feet, Ans. 2. If the hypothenuse of a right-angled triangle is 75 feet, and one of the other sides 60 feet, what is the third side?

752-602

OPERATION.

5625 · 3600 = 2025; √2025 = 45, feet, Ans.

3. A fort which is 15 feet high is surrounded by a moat 20 feet wide; what must be the length of a ladder that will just reach from the outer edge of the moat to the top of the fort? Ans. 25 feet.

4. Two men travel from the same place, one due east, and the other due north. One travels the first day 60 miles, and the other 80 miles. How far apart are they at the end of the day? Ans. 100 miles.

5. A line 36 meters long will exactly reach from the top of a perpendicular tower standing on the brink of a river, known to be 24 meters broad, to the opposite bank; what is the hight of the tower? Ans. 26.83+ meters.

6. A tree broken off 30 feet from the ground and resting on the stump, touches the ground 40 feet from the stump; what was the hight of the tree? Ans. 80 feet.

7. The rafters of a house, each 25 feet long, meet at the edge of the roof 15 feet above the attic floor; required the width of the house. Ans. 40 feet.

What is the second Principle? The third?