723. To find the rate of interest paid by a bor.

rower.

1. A buys a loan on 10 shares, Net Plan, at the beginning of a series, at $60 premium per share, and pays $10 dues and $7 interest on net sum received, for 8 years; what is the average or equated rate of interest?

years at 6%, or $429.25; hence the actual cost of the loan is $1700 +$429.25, or $2129.25; therefore

$2129.25-$1400, or $729.25, is the interest on the loan for 8 years; and the interest for 1 year is $729.25÷8, or $87.51; hence the rate is $87.51-$1400.0625+ or 64%.

Rule.-I. Find the sum of the installments, and the interest on the installments for the equated time at 6%; their sum will be the entire cost of the loan.

II. Subtract the amount of the loan from its entire cost; the remainder will be the interest on the loan for the period, from which the rate is readily found by the method of simple interest.

WRITTEN EXERCISES.

2. Mr. Jay borrows $4600, at 56 cents premium a month, on the Installment Plan; what sum do his monthly payments aggregate, and what equated rate % will he pay if the series runs out in 9 years? Ans $58.88; 9.25%.

3. I buy a loan of 10 shares, new series, in an association on the Installment Plan, at 60 cents a month premium, and in another, a loan of 10 shares on the Gross Plan at $60 premium; what rate % do I pay for each loan if each series runs out in 8 years? Ans. Inst., 7.54%; Gross, 9.47%.

NOTE.-A more complete discussion of this subject will be found in Brooks's Higher Arithmetic.

SECTION XII.

MENSURATION.

724. Mensuration treats of the measurement of geo metrical magnitudes.

725. Geometrical Magnitudes consist of the Line, Surface, Volume, and Angle.

726. A Line is that which has length without breadth or thickness. Lines are either straight or curved.

727. A Straight Line is one that has the same direction at every point.

728. A Curved Line is one that changes its direction at every point. The word line used alone means a straight line.

729. Parallel Lines are those which have the samė direction. Parallel lines, it is thus seen, will never meet.

730. One line is said to be perpendicular to another when the adjacent angles formed by the two lines are equal. 731. An Angle is the opening between two lines which diverge from a common point.

732. A Right Angle is an angle formed by one line perpendicular to another; as, ABC.

733. An Acute Angle is an angle less than a right angle; as, DEF. An Obtuse Angle is one larger than a right angle; as, DEG.

B

D

G

E

MENSURATION OF SURFACES.

734. A Surface is that which has length and breadth without thickness. Surfaces are plane or curved.

735. A Plane Surface is a surface such that if any two

of its points be joined by a straight line, every part of that line will lie in the surface.

736. A Plane Figure is a plane surface bounded by lines, either straight or curved

737. A Polygon is a figure bounded by straight lines; as, ABCDF.

A Polygon

of three sides is called a Triangle, of four sides, a Quadrilateral, etc.

E

A

B

738. A Diagonal of a polygon is a line ja ning the vertices of two angles not consecutive.

739. The Perimeter of a polygon is the am of its

sides.

740. The Area of a plane figure is the number of square units in its surface.

NOTE. The principles of mensuration are derived from geometry; their application to practical purposes is usually given in arithmetic.

THE TRIANGLE.

741. A Triangle is a polygon of three sides and three angles; as, ABC.

742. The Base is the side upon whicn it seems to stand; as, AB. The Altitude is a line perpendicular to the base, drawn from the angle opposite; as, CD.

B

D

743. An Equilateral Triangle is a triangle which has its three sides equal; when two sides are equal it is called isosceles; when its sides are unequal it is called scalene.

Rule. To find the area of a triangle, multiply the base by one-half of the altitude.

NOTE.-If the three sides are given and not the altitude, take haif the sum of the sides, subtract from it each side separately, multiply the half sum and these remainders together, and take the square root of the product.

1. What is the area of a triangle whose base is 25 rods and altitude 18 rods? Ans. 225 sq. rd., or 1 A. 65 P.

2. Required the area of a triangle whose base is 75 rods and altitude 57 rods dus. 13 A. 57P.

3. Required the area of a triangular field whose base is 965 rods and altitude 576 rods. Ans. 1737 A.

4. What is the area of a field whose sides are respectively 20, 30, and 40 chains? Ans. 29 A. 8 P.—.

THE QUADRILATERAL.

744. A Quadrilateral is a polygon having four sides and therefore four angles There are three classes, the par.

allelogram, trapezoid, and trapezium.

745. A Parallelogram is a quadrilateral whose opposite sides are parallel. The altitude is the perpendicular distance between its opposite sides.

746. A parallelogram which is right-angled

is called a Rectangle. When the four sides are equal it is called a Square.

747. An oblique-angled parallelogram

is called a Rhomboid. An equilateral rhomboid is called a Rhombus.

Rule. To find the area of a parallelogram, multiply the base by the altitude.

1. What is the area of a parallelogram 20 feet long and 18 feet wide? Ans. 40 sq. yd.

2. A has a rectangular lot 192 chains long and 65 chains wide; what is its area? Ans. 1248 acres.

3. What is the difference in the area of two lots, one being 245 rd. long, 42 rd. wide, and the other 85 chains long and 18 chains wide? Ans. 88 A. 110 P.

748. A Trapezoid is a quadrilateral which has two of its sides parallel. Its altitude is the perpendicular distance between its parallel sides.

Rule. To find the area of a trapezoid, multiply onehalf the sum of the parallel sides by the altitude.

1. Required the area of a trapezoid, one side being 120 in., the other 96 in., and the altitude 48 in. Ans. 36 sq. feet. 2. What is the area of a trapezoid, the sides being 365 and 124 in., and the altitude 86 in. ? Ans 146 sq. ft. 3 sq. in

3. What is the area of a plank 12 feet long, 18 inches wide at one end, and 12 inches at the other end?

Ans. 15 sq. ft.

4. A farmer has a field in the form of a trapezoid, the two parallel sides being 95 and 75 rods respectively, and the perpendicular distance between them being 65 rods; how much land in the field? Ans. 34 A. 85 P.

749. A Trapezium is a quadrilateral which has none of its sides parallel. A diagonal, as AB, divides the trapezium into two triangles.

A

B

Rule. To find the area of a trapezium, divide the trape zium into two triangles by a diagonal, find the area of each triangle and take the sum.

1. What is the area of a trapezium whose diagonal is 145 in., and the altitudes of the triangles, the diagonal being the base, are 30 and 40 inches respectively?

Ans. 35 sq. ft. 35 sq. in. 2. Required the area of a trapezium, the length of whose sides are respectively 20, 30, 25, and 35 chains, and the length of the diagonal 40 chains. Ans. 72 A. 56 P.—. ·

THE CIRCLE.

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

751. The curved line is called the cir

cumference, and a line passing through the centre and ending in the circumference is the diameter. Half the diameter is called the radius.

752. Rule. To find the circumference of a circle, mul tiply the diameter by 3.1416.

1. What is the circumference of a circle whose diameter is 25 inches? Ans. 78.54 in.