3. The first term of a geometrical progression is 10, the ratio 4, and the number of terms 6. What is the 6th term? 4. If a farmer should hire a man for 10 days, giving him 5 cents for the first day, 3 times that sum for the second day, and so on, what would be his wages for the last day?

5. If the first term is $100 and the ratio 1.06, what is the 6th term? Or, what is the amount of $100 at compound interest for 5 years at 6%?

6. What is the amount of $520 for 6 years, at 5% compound interest?

7. What is the sum of a geometrical series, of which the first term is 5, the ratio 3, and the number of terms 5?

RULE.-The sum of a geometrical series is equal to the difference between the first term, and the product of the last term by the ratio, divided by the difference between the ratio and 1.

8. The extremes of a geometrical progression are 4 and 1024, and the ratio 4. What is the sum of the series? 9. The extremes are is the sum of the series?

and 34 and the ratio 23. What

10. What is the sum of the series in which the first term is 2, the last term 0, and the ratio ; or what is the sum of the infinite series 2, 1, 1, 1, 1, 1, 2, etc?

MENSURATION

DEFINITIONS.

545. Mensuration treats of the measurement of lines, surfaces, and solids.

546. A Line is that which has length only.

547. A Straight Line is a line that does not change its direction.

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

549. Parallel Lines are such as are equidistant throughout their whole extent.

Straight Line.

Curved Lines.

Parallel Lines.

550. A Plane Surface is a surface such that a straight line joining any two points of it is wholly in the surface.

551. A Curved Surface is a surface such that no part of it is a plane surface.

552. An Angle is the divergence of

two lines that meet.

553. A Right Angle is the angle formed when one straight line meets another making the adjacent angles equal.

The lines are perpendicular to each other when a right angle is formed.

Angle.

Two Right Angles.

(341)

Acute Angle,

Obtuse Anglo.

Triangle.

Quadrilateral.

Parallelogram.

Rectangle,

Polygon.

Circumference

Diameter

Radius

Circle.

554. An Acute Angle is an angle which is less than a right angle.

555. An Obtuse Angle is an angle which is greater than a right angle.

556. The Vertex of an angle is the point where the sides meet.

557. A Triangle is a figure which has three sides and three angles.

558. A Quadrilateral is a figure bounded by four sides.

559. A Parallelogram is a quadrilateral whose opposite sides are parallel.

560. A Rectangle is a parallelogram whose angles are right angles.

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

562. A Circle is a plane figure bounded by a curved line every point of which is equally distant from a point within called the center.

563. The Circumference is the line which bounds the circle.

564. A Radius of a circle is a straight line drawn from the center to the circumference.

565. A Diameter of a circle is a straight line drawn through the center and terminating at both ends in the circumfer

ence.

566. The Base of a figure is the side. on which it is assumed to stand.

567. The Altitude of a figure is the perpendicular distance between the base and the highest point opposite it.

568. A Diagonal of a figure is a straight line joining the vertices of two angles not adjacent.

569. The Perimeter of a figure is the length of the lines that bound it.

570. The Area of a surface is the definite amount of surface it contains.

MEASUREMENT OF LINES.

571. It can be shown by geometry that the circumference of a circle is 3.1416 times its diameter.

For ordinary measurements it is sufficiently accurate to consider the circumference 34 times the diameter.

RULE.-1. The circumference is equal to the diameter multiplied by 3.1416.

2. The circumference divided by 3.1416 is equal to the diameter.

WRITTEN EXERCISES.

572. 1. What is the circumference of a circle 10 feet in diameter?

2. What is the circumference of a circle 45 feet in diameter? 3. How far is it around a circular lake that is 300 rods in diameter ?

4. What is the circumference of a circle whose radius is 20 rods?

5. What is the circumference of a circle whose radius is 5 feet 6 inches?

6. What is the diameter of a circle whose circumference is 318.5 rods?

7. What is the radius of a circle whose circumference is 1284 rods?

MEASUREMENT OF SURFACES.

573. To compute the area of a parallelogram. The truth of the following principle has been shown already: PRINCIPLE.-The area of any rectangular figure is equal to .the product of its length by its breadth or altitude.

D

Altitude

BF

By examining the figure A, B, C, D, it will be seen that it is cqual to E, F, D, C, and that any oblique parallelogram is equal to a rectangu lar parallelogram of the same base and altitude. Therefore,

RULE.-The area of any parallelogram is equal to the prod uct of the base multiplied by the altitude.

WRITTEN EXERCISES.

574. 1. How many square feet are there in a parallelogram, whose length is 40 feet and altitude 13 feet?

2. What is the area of a parallelogram whose base measures 7 feet and whose altitude is 3 feet 8 inches?

3. What is the area of a field in the form of a parallelogram, whose length is 30 rods and the perpendicular distance between the sides is 24 rods?

4. What is the area of a parallelogram whose length is 35 feet and whose altitude is 15 feet?