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6. Required the plan and area of a piece of ground from the

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7. Required the plan and area of a field from the following equi

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Answer.

727 The first and last ordinates. 4384 Four times the sum, &c.

1540 Twice the sum, &c.

6651 Sum total.

100 The common distance.

3)665100

221700

36288 Trapezoid at the end.

2.57988 Area in square links.

4

2.31952

40

12.78080 Area 2a. 2r. 123p.

NOTE. Whenever the Rule given in this Problem can be applied, it will be found more easy, expeditious, and accurate, in finding the areas of offsets, and of narrow pieces of land, than the Rules for triangles and trapezoids. (See my Mensuration, page 222.)

PROBLEM X.

To find the Breadth of a River.

EXAMPLE.

Let the following figure represent a river, the breadth of which is required.

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Fix upon any object B, close by the edge of the river, on the side opposite to which you stand. By the help of your cross, make A D perpendicular to A B; also make A C C D, and erect the perpendicular D E; and when you have arrived at the point E, in a direct line with C B, the distance D E will be A B, the breadth of the river; for by Theo. 1. Part I. the angle A C B = DC E, and as A C C D, and the angles A and D are right angles, it is evident that the triangles A B C, C D E are not only similar but equal.

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NOTE 1. The distance between A and the edge of the river, must be deduced from D E, when it is not convenient to fix A close by the river's edge.

2. This Problem may also be well applied in measuring the distance of an inaccessible object; for let A C equal 8, C D equal 2, and D E equal 10 chains; then, by similar triangles, as CD: DE:: AC: A B; that is, as 2: 10::8: 40 chains = A B. (See Theo. 11. Part I.)

PROBLEM XI.

Lines upon which there are Impediments not obstructing the Sight.

Suppose m n, to represent a deep pit or water, and A and B two objects, the direct distance of which is required.

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PROBLEM XII.

Lines upon which there are Impediments obstructing the Sight.

EXAMPLE.

Suppose C D E F to represent the base of a building, which obstructs the sight, and through which it is necessary that a straight line should pass from an object

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PART THE FOURTH.

THE METHOD OF SURVEYING WITH THE CHAIN ONLY; AND OF MEASURING MERES, WOODS, DISTANCES, LINES UPON WHICH THERE ARE IMPEDIMENTS, AND HILLY GROUND.

MISCELLANEOUS INSTRUCTIONS.

THE method of surveying with the chain only, is adopted by most Practical Surveyors, and is certainly preferable to that by the chain and cross; because it is not only as accurate, but generally more expeditious.

Whatever be the form of the field or ground to be surveyed, you must measure as many lines as will enable you to plot it with accuracy. The plan being drawn, you may then divide the figure into trapeziums, triangles, &c.; and measure the diagonals, perpendiculars, &c. with your plotting-scale.

It is better, however, to divide small pieces, and single fields, into trapeziums and triangles, by measuring the diagonals and bases during the survey; so that to find the area, you will have only the perpendiculars to measure with the scale.

You must also measure, in some convenient direction, a proof-line to each trapezium and triangle.

NOTE 1. The offsets must be treated according to the directions in Part III. Prob. 6. Or, you may reduce the crooked sides to straight ones, by including as much of what does not belong to the field under your survey, as you exclude of what does, in the following manner. Apply to the crooked line in question, the straight edge of a clear piece of lantern horn, so that the small parts cut off by it, from the crooked figure, may be equal to those which are taken in; (of this equality you will presently be able to judge very correctly, by a little practice;) then, with a pencil, draw a line by the edge of the horn. The sides being thus successively straightened, the content may be easily found.

2. A slender bow of cane or whale-bone, strung with a silk thread, may be substituted for the horn. The thread must be applied to the crooked fence, and two marks made, by which a straight line must be drawn.

3. The sides may also be straightened by a parallel ruler; but the operation is generally tedious, and must be performed with the greatest care, or it will not be more correct than the foregoing method.

4. When the three sides of a triangle are given, the area may be found as follows.

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