EXAMPLE 2. What is the area of a trapezoid, the parallel sides of which are 14 chains 20 links, and 12 chains 35 links, and the distance between them 27 chains 25 links?

A. R. P.
Answer 36. 0. 28.

EXAMPLE 3. Required the area of a trapezoidal field whose sides are 40 chains and 27 chains, and distance 15 chains and a half.

A. R. P.

Answer 51. 3. 28.

PROBLEM V.

To find the area of a parallelogram, whose angles are not

EXAMPLE 1. Required the area of a four-sided regular field, whose sides are 20 chains 15 links, and 16 chains 89 links, and the included angle 30 degrees.

Plot the figure to any scale; its perpendicular height will then be found to be 8.45 chains,

and 20.15 x 8.45 170.265 square chains.

A. R. P.

Answer 17. 0. 4.

EXAMPLE 2. When the sides are 15 chains, 25 links; and 21 chains 18; and the included angle 45 degrees; what is the area of the field?

A. R. P.

Answer 22. 2. 26.

EXAMPLE 3. With the same sides, but with an angle of 105 degrees, what is the area?

A. R. P.
Answer 31. 0. 31.

PROBLEM VI.

To find the area of a trapezium.

Rule. Divide the trapezium into two triangles by the longest diagonal (see Geometrical Problems, 13th Ex.); take this as the common base of the two triangles, and multiply it by half the sum of the two perpendiculars (BE and FD in the diagram, let fall upon it, from the opposite angles).

EXAMPLE 1. To find the area of a trapezium, whose diagonal is 20 chains, and the two perpendiculars 2 chains 50 links, and 3 chains 40 links.

2.50

3:40

25.90

A. Ꭱ. P.

Answer 2.95 x 20-59 sq. chains 5. 3. 24.

EXAMPLE 2. How many square feet of paving are there in a court yard, whose diagonal is 2 chains 64 links, and perpendiculars 95 links and 84 links?

Answer 10292 square feet.

PROBLEM VII.

To find the area of an irregular polygon.

Rule. Divide the polygon into trapeziums and triangles, and find the sum of the areas of each.

Various examples of this will be found in the part on chain surveying.

PROBLEM VIII.

To find the circumference of a circle.

Rule. Multiply the diameter by 3.1416.

EXAMPLE 1. What is the circumference of a circle, whose diameter is 30 chains?

Circumference = 3∙1416 × 30 94.2480 chains.

EXAMPLE 2. Required the circumference of a circle, whose diameter is 17 chains 40 links.

Answer 54.66 chains.

EXAMPLE 3. What is the diameter of a circle, whose circumference measures 1,000 chains?

Answer 318 chains 30 links.

PROBLEM IX.

To find the length of any arc.

As 360°: to the given degrees of the arc :: the whole circumference: to the length of the arc required.

EXAMPLE 1. What is the length of an arc of 20°, of a circle, whose circumference measures 850 chains?

As 360°: 20°:: 850 chains : x

EXAMPLE 2. Required the length of the quadrant, the circumference measuring 300 chains.

Answer 75 chains.

EXAMPLE 3. What is the circumference of a circle, when the arc of 30° measures 17 chains 20 links?

Answer 206.40 chains.

PROBLEM X.

To find the area of a circle.

Rule. Multiply the square of the diameter by 7854.

EXAMPLE 1. What is the area of a circular field, whose diameter is 18 chains?

18o × 7854 = 254-4696 sq. chains 25·44696 acres.

A. R. P. Answer 25. 1. 31.

EXAMPLE 2. The area of a circular plot of ground is required, whose diameter is 27 chains.

A. R. P.

Answer 57. 1. 1.

EXAMPLE 3. What is the area of the circle, that can be described by a rope, measuring 5 chains, having one end fixed, as a centre?

A. R. P.

Answer 9. 2. 0.

CHAP. I.

ON THE CHAIN.

THE chain, in common use, is called Gunter's chain, from its inventor, and is divided into ten equal parts, distinguished by a piece of brass, with notches; the brass at the first division, from either end, having one notch; at the second division, two notches; at the third, three; at the fourth, four; and at 50 links, or the middle of the chain, there is a round piece of brass.

The object of marking these divisions from either end of the chain, is to enable the surveyor to measure either way from each end.

Each of the above brass divisions of the chain is again sub-divided into other ten parts, or links; so that the whole chain is divided into 100 parts, or links; each link therefore chs001 chs.; and each 10 links chs=0·10 chs. The advantage of this arrangement is, that, in measuring a line, it matters not whether the distance be termed 7.32 chains (7 chains 32 links,) or 732 links; and, as 10 square chains make one acre, or 10 times (100 links) or 10 (10000 links) or 100000 square links, it is only necessary to multiply together the length and breadth, given in links, of a piece of ground, whose area is required, and set off four decimal places, when the integers will be square chains; or set off five decimal places, and the integers will be the required acres.