(108) Chain Perpendiculars. Perpendiculars may be set out with the chain alone, by a variety of methods. These methods generally consist in performing on the ground, the operations executed on paper in practical geometry, the chain being used, in the place of the compasses, to describe the necessary arcs.

As these operations, however, are less often used for the method of surveying now to be explained, than for overcoming obstacles to measurement, it will be more convenient to consider them in that connection, in Chapter V.

DIAGONALS AND PERPENDICULARS.

(109) In Chapter I, of this Part, we have seen that plats of surveys made with the chain alone, have their contents most easily determined by measuring, on the plat, the perpendiculars of each of the triangles, into which the diagonals measured on the ground have divided the field. In the Method of Surveying by Diagonals and Perpendiculars, now to be explained, the perpendiculars are measured on the ground. The content of the field can, therefore, be found at once, (by adding together the half products of each perpendicular by the diagonal on which it is let fall,) without the necessity of previously making a plat, or of measuring the sides of the field. This is, therefore, the most rapid and easy method of surveying when the content alone is required, and is particularly applicable to the measurement of the ground occupied by crops, for the purpose of determining the number of bushels grown to the acre, the amount to be paid for mowing by the acre, &c.

(110) A three-sided field. Measure the

Fig. 63.

longest side, as AB, and the perpendicular, CD, let fall on it from the opposite angle C. Then the content is equal to half the product

A

D

of the side by the perpendicular. If obsta- E с

B

cles prevent this, find the point, where a perpendicular let fall from an angle, as A, to the opposite side produced, as BC, would meet

it, as at E in the figure. Then half the product of AE by CB is the content of the triangle.

A

Fig. 64.

B

1.10

2.00

2.80

-4.80 C

1.75

D

(111) A four-sided field. Measure the diagonal AC. Leave marks at the points on this diagonal at which perpendiculars from B and from D would meet it; finding these points by trial, as previously directed in Arts. (104) and (107). The best marks at these "False Stations," have been described in Art. (90). Return to these false stations and measure the perpendiculars. When these perpendiculars are measured before finishing the measurement of the diagonal, great care is necessary to avoid making mistakes in the length of the diagonal, when the chainmen return to continue its measurement. One check is to leave at the mark as many pins as have been taken up by the hind-chainman in coming to that point from the beginning of the line.

Example 9. Required the content of the field of Fig. 64. Ans. 0A. 2R. 29P. The field may be platted from these measurements, if desired, but with more liability to inaccuracy than in the first method, in which the sides are measured. The plat of the figure is 3 chains to 1 inch.

The field-notes may be taken by writing the measurements on a sketch, as in the figure; or in more complicated cases, by the column method, as below. A new symbol may be employed, this mark,, or, to show the False Station, from which a perpendicular is to be measured.

(112) A many-sided field. Fig. 65, and the accompanying field-notes represent the field which was surveyed by the First Method and platted in Fig. 51.

(113) A small field which has many sides, may sometimes be conveniently surveyed by taking one diagonal and measuring the perpendiculars let fall on it from each angle of the field, and thus dividing the whole area into triangles and trapezoids; as in Fig. 36, page 48.

The line on which the perpendiculars are to be let fall, may also be outside of the field, as in Fig. 37, page 48.

Such a survey can be platted very readily, but the length of the perpendiculars renders the plat less accurate.

This procedure supplies a transition to the method of " Offsets," which is explained in the next article.

OFFSETS.

(114) Offsets are short perpendiculars, measured from a straight line, to the angles of a crooked or zigzag line, near which the straight

A

1.00

C

Fig. 66.

D

1.50

50

B

line runs. Thus, in the figure, let ACDB be a crooked fence, bounding one side of a field. Chain along the straight line AB, which runs from one end of the fence to the other, and, when opposite each corner, note the distance from the beginning, or the point A, and also measure and note the perpendicular distance of each corner C and D from the line. These corners will then be "determined" by the Second Method, Art. (6).

The Field-notes, corresponding to Fig. 66, are as in the margin. The measurements along the line are written in the column, as before, counting from the beginning of the line, and the offsets are written beside it, on the right or left, opposite the distance at which they are taken.

0300 to B.

D 25 250

C 30 100

A sketch of the crooked line is also usually From A 0 O

made in the Field-notes, though not abso

lutely necessary in so simple a case as

this. The letters C and D would not be used in practice, but are

here inserted to show the connection between the Field-notes and

the plat.

In taking the Field-Notes, the widths of the offsets should not be drawn proportionally to the distances between them, but the breadths should be greatly exaggerated in proportion to the lengths.

(115) A more extended example, with a little different notation, is given below. In the figure, which is on a scale of 8 chains to one inch for the distances along the line, the breadths of the offsets are exaggerated to four times their true proportional dimensions.

(116) The plat and Field-notes of the position of two houses, determined by offsets, are given below, on a scale of 2 chains to 1

(117) Double offsets are sometimes convenient; and sometimes triple and quadruple ones. Below are given the notes and the plat, 1 chain to 1 inch, of a road of varying width, both sides of which are determined by double offsets. It will be seen that the line AB crosses one side of the road at 160 links from A, and the other side of it at 220.