A Treatise on Practical Mensuration in Eight Parts ...

continue the line to +6. From + 6, proceed as before, until you arrive at +14; and you will have obtained the following dimensions, from which a plan may be drawn.

2. Let the foregoing figure represent a river, a plan of which is required.

Begin at a, and measure to c; taking offsets to the river's edge, as you proceed. From c measure to d; and there take the tie or chord-line db, which will enable you to lay down the first and second lines. Continue the second line to n; and from m, measure to r, at which place take the tie-line rn; and thus proceed until you come to the end of your survey at x.

If the breadth of the river be every where nearly the same, its breadth taken in different places, by the next Problem, will suffice; but if it be very irregular, di◄ mensions must be taken on both sides, as above.

When the area is required, it must be found from the plan, by dividing the river into several parts; and taking the necessary dimensions by the scale.

Note 1. Any bog, marsh, mere, or wood, whatever may be its number of sides, may be measured by this Problem.

2. In taking an angle with the chain, some surveyors never measure more than 100 links from the angular point but when the lines, including the angle, are of a considerable length, it is much better tomeasure the chord-line at a greater distance from the angle, than one chain; because a small inaccuracy in constructing the figure, when the angular distance is short, will throw the lines, when far produced, considerably out of their true position. It some times happens, however, in consequence of obstructions, that it is impossible to measure the chord-line at greater distance from the angular point, than one chain. In such cases, multiply both the angular distance and chord-line by 2, 3, 4, or any larger number, as circumstances may require; and use the products resulting, in laying down the angle. Or, which is the same thing, lay down the angle by a scale 2 or 3 times as large as that by which you intend to draw the plan. By this means the radius or angular distance be equal cd, and the chord-line bd, will be increased in length; and consequently the lines ac and en, including the angle, will be more correctly laid down in position.

PROBLEM VII.

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 ri ver, on the side opposite to which you stand. By the help of your cross, make AD perpendicular to AB; also make ACCD, and erect the perpendicular DE; and when you have arrived at the point E, in a direct line with CB, the distance DE will be = AB, the breadth of the river; for by Theo. 1, Part I, the angle ACB DCE, and as AC CD, and the angles A and D right angles, it is evident that the triangles ABC, CDE are not only similar but equal.

Note, 1. The distance between A and the edge of the river must be deducted from DE, 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 any inaccessible object: for let AC equal 8, CD equal 2, and DE equal 10 chains; then, by similar triangles, as CD: DE:: AC: AB equal to 40 chains.

PROBLEM VIII.

To measure a line upon which there is an impediment.

EXAMPLE.

Let CDEF represent the base of a building, through which it is necessary that a straight line should pass from A.'

Measure from A to m; at m, erect the perpendicular ma, which measure until you are clear of the impediment, as at c. Erect the perpendicular ce, which measure until you are beyond the building, as at b. Erect the perpendicular bd; and make bn equal to me, at which point you will be in a direct line with mA. Erect the perpendicular nB, which measure; then Am added to the sum of cb and nB, will give the length of the whole line AB.

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Note 1. This Problem is very useful when you meet with ponds, bogs, buildings, &c. upon a chain-line.

2. When a cross is not at hand, a perpendicular may be erected by the chain, in the following manner: Measure 40 links from m to r; then let one end of the chain be kept fast at r, and the eightieth link at m; take hold of the fiftieth link, and stretch the chain so that the two parts mw, and rw, may be equally tight; then will be mw be perpendicular to mr, for mw, mr, and ware in proportion to each other as 3, 4, and 5. (See Problem 29, Part I.).

PROBLEM IX.

To part from a rectangular field, any proposed quantity of land, by a line parallel to one of its sides.

RULE.

Divide the proposed area, în square links, by the side upon which it is to be parted off, and the quotient will be length of the other side of the figure required.

Note 1. If there be offsets on the line upon which the proposed quantity is to be parted off, deduct the area of the offsets from it, and proceed with the remainder as above.

2. Acres, roods, and perches may be reduced into square links, by multiplying the whole quantity, in perches, by 625, the number of square links in a perch.

EXAMPLES.

...1. From the rectangle ABCD part off 2 acres, 1 rood, and 52 perches, by a line parallel to AD = 700 links,