SHORT METHOD OF CUBE ROOT.

407. The following method of cube root is the shortest known. The shortness consists in obtaining the successive trial divisors by using the previous work.

21. What is the cube root of 12895213625 ?

EXPLANATION.-The process is the same as in the previous method up to finding the second trial divisor.

To find the second trial divisor, 1587, we add the square of the last figure of the root (9), the complete divisor (1389), the second correction (9), and the first correction (18), as indicated by the bracket. In the same manner each succeeding trial divisor is obtained.

RULE.-Find the first trial divisor by the usual method.

Find each succeeding trial divisor by adding the square of the last figure of the root to the last complete divisor and the two corrections required to obtain it, and proceed as before.

This method is merely a modified form of the previous method, and is readily explained by the algebraic formula. It is the labor saving method of cube root, and should be thoroughly mastered by the student. Extract the cube root of the following:

22. 41063625. 23. 347428927. 24. 410172407.

25.

633839779.

26. 1879080904.
27. 95256152263.

28. 196426.902797. 29. 436.036824287. 30. .000890277128.

PRACTICAL MEASUREMENTS.

LINES.

408. A Line is that which has length only. 409. A Straight Line is a line that has the same direction at every point.

410. A Curved Line is a line that changes its direction at every point.

The word line is used for straight line, and curve for curved line.

411. Parallel Lines are lines that have the same direction at every point and remain equidistant.

ANGLES.

412. An Angle is the difference in direction of two lines that meet.

The point in which the lines meet is called the Vertex, and the lines are called the sides of the angle.

413. A Right Angle is an angle formed by one line perpendicular to another.

414. An Acute Angle is an angle smaller than a right angle.

ACUTE ANGLE

415. An Obtuse Angle is an angle larger than a right angle.

-K

H

OBTUSE ANGLE

SURFACES.

416. A Surface is that which has length and breadth only.

417. A Plane Surface, or a Plane, is a surface in which if any two points are taken, the straight line joining these points will lie wholly in the surface.

418. A Plane Figure is a plane bounded by lines either straight or curved.

419. A Polygon is a plane figure bounded by straight lines.

The Sides of a polygon are the lines which bound it; and the Perimeter is the sum of the sides, or the distance around it.

E

A

D

POLYGON

C

420. A Polygon of three sides is called a triangle; of four sides, a quadrilateral; of five sides, a pentagon; of six sides, a hexagon; of seven sides, a heptagon, of eight sides, an octagon, etc.

421. a. An Equilateral Polygon is one whose sides are equal. b. An Equiangular Polygon is one whose angles are equal.

422. A Diagonal of a polygon is a line joining the vertices of any two angles not adjacent.

423. The Area of a polygon is the number of square units in its

surface.

THE RECTANGLE.

424. A Rectangle is a polygon having four sides and four right angles.

When the word side is used straight side is meant.

A Square is a rectangle whose sides are equal.

The Dimensions of a rectangle are its length and breadth.

The Base of a rectangle is the side upon which it seems to stand. The side opposite is called the Upper Base.

The Altitude of a rectangle is the perpendicular distance be tween its bases.

425. To find the area of a Rectangle.

In the rectangle ABCD, the area is the number of square units it contains, which is the number in each row multiplied by the number of rows; but this is equal to the number of linear units in the length multiplied by the number in the breadth. Hence,

D

A

B

RULE.-Multiply the length by the breadth.

1. The dimensions must be expressed in uni s of the same denomination. 2. To find either side of a rectangle, divide the area by the other side.

EXAMPLES.

426. 1. How many square feet in a floor 25 feet long and 20 feet wide ?

2. How many square rods in a field 40 rods long and 18 rods wide ?

3. How many acres are there in a square farm each of whose sides is 120 rods?

4. What is the width of a field 60 rods long which contains 15 acres?

5. A field is 16 chains long and 30 rods wide. How many acres does it contain?

6. A certain garden is 150 feet long and 100 feet wide. Find its area in square meters.

7. Mr. G. has a rectangular field 40 decameters long and 35 dekameters wide. How many hectares does the field contain?

427. To find the area of a Parallelogram.

428. A Parallelogram is a quadrilateral having its opposite sides parallel.

If the parallelogram ABCD be divided by the line CE, and the triangle AEC be placed at the right, we shall have the rectangle EFCD, whose base, altitude and area is the same as that of the parallelogram. Hence,

RULE.-Multiply the base by the altitude.

ALTITUDE

A

E

B

D

EXAMPLES.

429. 1. What is the area of a parallelogram whose base is 24 feet and altitude 15 feet?

2. Find the area of a parallelogram whose base is 15 yards and altitude 18 feet.

3. A field in the form of a parallelogram contains 3 acres. Its length is 24 rods. What is its width, or altitude?

THE TRIANGLE.

430. A Triangle is a polygon having three sides and three angles.

The Altitude of a triangle is the perpendicular distance from the vertex to the base, or the base produced or lengthened out, as CD.

A

B

A Scalene triangle is one whose sides are all unequal; an Isosceles triangle is one two of whose sides are equal; an Equilateral triangle is one whose three sides are equal.

A Right triangle is one which has a right angle; an Obtuse triangle is one which has an obtuse angle; an Acute triangle is one whose angles are all acute an Equiangular triangle is one whose three angles are equal.

431. To find the area of a Triangle when the Base and Altitude are given.

The triangle ABC is seen to be one-half of the parallelogram ABCD; but each has the same base AB and the same altitude CE. Hence,

C

A

E

RULE.-Multiply the base by one-half of the altitude.

EXAMPLES.

432. 1. What is the area of a triangle whose base is 40 feet and altitude 24 feet?

2. What is the area of a triangle whose base is 42 feet and altitude 19 feet?

3. How many acres in a triangular field whose base is 60 rods and altitude 48 rods ?

4. How many square feet of boards will be required to cover the gables of a house that is 32 feet wide, the ridge of the roof being 11 feet above the square?

433. To find the area of a Triangle when the Three Sides are given.

The principles of practical measurements are derived from Geometry. The rules founded upon these principles cannot always be clearly illus trated or explained by arithmetic. In such cases only the rule with examples will be given.