Proceeding on to G, a distance of 320 yds. from B, it is found that the perpendicular CG is 160 yds., and then at н, a distance of 420 yds. from B, it is found that the perpendicular EH is 90 yds. Find the area.

The diagonal CE is 312 yds., and its perpendicular KD is 115 yds. The side BC is 125 yds.

Definitions. The area of a long irregular figure can best be found by means of ordinates or offsets. Take any

straight line AB, and at points A, E, F, G, &c, let AD, Ee, Ff, &c., be drawn perpendicular to AB. Then these perpendiculars AD, Ee, Ff, &c., are called ordinates or offsets.

The offsets may be taken at equal distances AE, EF, FG, &c., as in fig. 1; and also at unequal distances, as in fig. 2, in which case the whole figure is divided into triangles and trapezoids.

RULES. To find the area of a long irregular figure.

(1) When the offsets are taken at equal distances (fig. 1). To half the sum of the first and last breadths, add all the intermediate breadths; divide the sum by the number of equal parts in the line AB (and not by the number of breadths), and the quotient is the mean breadth of the figure. Multiply the mean breadth by the length of the figure, and the product is the area, nearly.

(2) When the offsets are taken at unequal distances (fig. 2).—Find the areas of all the triangles and trapezoids into which the figure is divided; and their sum is the area, very nearly.

Note 1.-Another rule also may be given, which can be used in cases where great accuracy is not essential.-Add all the breadths together, and divide by the number of them for a mean breadth; multiply this mean breadth by the length of the figure, and the product is the area, nearly.

Note 2.-The rules in this Section will give the area a little less than it actually is, but sufficiently correct for all practical purposes; while the rule given in Note 1 gives the area more than it really is, and with less exactness than the other.

Note 3.-All the questions in this Section may be worked either by Rule 1 or Rule 2: the first rule, however, is generally employed when the ordinates are taken at equal distances; but when they are taken at unequal distances, it will be better to employ Rule 2.

Example 1.—The perpendicular breadths or offsets of any irregular figure (fig. 1) at five equidistant places A, E, F, &c., are 10 ft., 7 ft., 9 ft., 6 ft., and 8 ft., and its length is 30 ft.; find the area.

9 =half the sum of extreme breadths 7)

9 =intermediate breadths

6

4)31

7.75 mean breadth

30

232.5 sq. ft. area, nearly,

Example 2.-Find the area of an irregular plot of land (fig. 2) from the following offsets taken in yards :-At 12, 10; at 30, 16; at 40, 24; at 64, 16; at 84, 20.

Here the figure consists of one triangle and four trapezoids; hence the area of the figure is the sum of the areas of the triangle and trapezoids; and the question may be thus worked :—

12 × 10=120 18x (10+16)=468

10x (16+24)=400

24 × (24+16)=960

20 × (16+20)=720

2)2668

1334 square yards—area.

EXAMPLES.

[Questions 1 and 2 are worked by Rule 1; all the rest by Rule 2.]

(1) Find the area of an irregular figure, when its length is 60 ft., and its breadths, taken at six equidistant places, are 24, 14, 16, 16, 18, and 22 ft.

(2) Find the area of an irregular plot of land, which is 40 yds. long, and its breadths, taken at six equidistant places, are 0, 4, 10, 16, 24, and 8 yds.

(3) Find the area of a plot of land from the following offsets, taken in yards :-At 12, 20; at 20, 18; at 50, 60; and at 80, 40.

(4) In measuring a plot of land, the following are the offsets taken in yards :-At 0, 60; at 20, 38; at 35, 30; at 60, 16; and at 70, 0. Find the area.

(5) In measuring a plot of land, the following offsets are taken in links:-At 0, 120; at 120, 164; at 200, 106; at 260, 60; and at 300, 0. Find the area.

(6) Find the area of an irregular plot of land when its offsets in yards are: at 0, 0; at 30, 20; at 50, 34; at 100, 26; at 140, 40; and at 180, 20.

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XI. THE CIRCLE.-THE CIRCUMFERENCE AND

DIAMETER.

Definitions.-A circle is a plane figure bounded by a curved line called the circumference or perimeter, and is such, that all lines drawn from the centre to the circumference are equal.

AB is the diameter of the circle. AO or BO is the radius of the circle.

B

RULES. (1) To find the circumference of a circle, when its diameter is given.

Multiply the diameter by 22; that is, multiply the diameter by 22, and divide the product by 7.

(2) To find the diameter of a circle, when the circumference is given.

Divide the circumference by 22; that is, multiply -the circumference by 7, and divide the product by 22.

Note 1.-The above rules will give the answer with sufficient accuracy for all practical purposes. The length of the circumference, found by multiplying the diameter by 22, will not be wrong to the one-hundredth part of the length of the radius. For instance, if the radius of a circle is 100 ft., then its circumference, obtained by multiplying the diameter by 22, will not be 1 ft. wrong.

Note 2.-If still greater accuracy is desirable, then, to find the circumference, multiply the diameter by 3·1416 or by ; and to find the diameter, divide the circumference by 3·1416 or by 3.

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