them;

draw a plot of the field, and find the area by PROBLEM XII.

FIELD BOOK. See Fig. 52.

Fig. 52

Draw the side AB, making it the given length 30.60; with the diagonal AC 45 and the side BC 20.40, describe cross arcs as at C, from the points A and B as centres, and the point of intersection will represent that corner of the field; draw the side BC and the dotted diagonal AC; with the dia gonal AD 35 and the side CD 22.40, describe cross arcs as at D, from the Points A and C, and draw the Side CD and the dotted diagonal AD. Proceed in this manner till all the sides and diagonals are drawn.

TO FIND THE AREA.

The field being plotted, may be divided into one trapezium and two triangles; the area of which is calcu ated as fol

As each of the sides of the several triangles, into which the preceding plot of a field is divided, is known from the field book, the area of the field may be calculated arithmetically, by finding the area of each triangle, according to PROB. IX. Rule 3; and then adding the whole together. This method, though it may require more time, is preferable to the other, because more accurate. Indeed it is always better to calculate the area of a field arithmetically than geometrically; for in the former no two persons can differ in their calculations; whereas, according to the latter, which is the common method of casting the contents of a field, it is hardly to be expected that any two persons will perfectly agree. The inaccuracy of scales, and the difficulty of determining with precision the length of sides and perpendiculars with a scale and dividers, render it almost if not quite impossible to obtain the exact area of a field, in the method commonly practised, even if the surveyor has measured it accurately in the first place.

Other methods of taking the survey of a field by the chain

are rather curious than useful; and it is much better to ascertain the angles by an accurate compass, or some instrument designed purposely for taking angles.

CASE IV.

TO SURVEY A FIELD WITH A CHAIN AND COMPASS.

Measure the length of the sides with a chain, and take their bearing or course with a compass ;* enter these in a field book; plot the field on paper, and calculate the area by the directions already given.

TO PROTRACT OR DRAW A MAP OF A FIELD.

Draw a line to represent a meridian, or north and south line, from which lay off a bearing or course of the first side of the field, with a protractor or from a line of chords; and from a scale of equal parts, measure the length of the side and draw a line to represent it.

At the end of this line draw a line parallel to the meridian line, and then lay off the second side of the field as before taught; proceed in the same manner to draw parallel lines, and lay off the several sides till the whole is protracted.

In protracting a field, let the top of the paper be considered. as north, the bottom south, the right hand east, and the left hand west; lay the course to the right or left of the meridian line, according as it is east or west, and upwards or downwards, according as it is north or south.

In all protractions, if the end of the last distance falls exactly on the point from which you began, the course also being right, the field work and protractions are truly taken and performed; if not, an error must have been committed in one of them in such cases, make a second protraction; if this agrees with the former, it is to be presumed the fault is in the field work; a re-survey must then be taken.

:

* A compass may be so constructed with two indexes, one moveable and the other fixed, as to ascertain the angle made by two sides, without reference to the bearing of those sides. Such a compass would be particularly useful in surveying land where there are mineral substances which have an influence upon the compass needle, attracting it one way or the other, and thus rendering it impossible to take a course by it

The sides of the several triangles into which the plot of a field is divided may be found by trigonometry; and then the area of each triangle may be calculated according to PROB. IX. Rule 3. The sum of the areas of the several triangles will be the area of the whole field. This method may require more time, but it is perfectly accurate, since no dependence is placed on the uncertain measurement of scale and dividers.*

In the preceding EXAMPLE, suppose the field divided into three triangles. See Fig. 53. In the triangle EAB, the sides EA and AB are known from the FIELD BOOK, and their contained angle is known from the bearing of the sides. The other angles and the side EB may be found by OBLIQUE TRIGONOMETRY, CASE III.; and then there will be the three sides to find the area. In the triangle EBC, the side BC is known from the FIELD BOOK, and the side EB is found as above mentioned; the angle EBA is also found as above; this subtracted from the angle ABC, which may be found from the bearing of the sides AB and BC, will leave the angle EBC : there will then be the two sides and their contained angle to find the third side; and this being found, there will be the three sides to find the area. In the triangle EDC, the sides DE and DC are known from the FIELD BOOK, and their contained angle is known from the bearing of the sides. The side EC and the area may be found as above.

It is recommended to the learner to make these calcula

*As accuracy is of much higher importance than expedition, no practical surveyor should ever depend upon his scale and dividers.-ED.

tions, as it will improve him in the knowledge of trigonometry.

NOTE. Two sides and their contained angle being given, the area may be found by PROB. IX. Rule 4.

ANOTHER METHOD OF PROTRACTING FIELDS.

Without drawing parallel lines at the end of each side, a field may be protracted by the angles made by the several sides; and the angle made between any two sides may be found by the following RULES.

RULE 1. If the course or bearing of one of the sides is north and the other south, one east and the other west, subtract the less course from the greater.

RULE 2. If one is north and the other south, and both east or west, add both courses together.

RULE 3. If both are north or south, and one east and the other west, subtract their sum from 180°.

RULE 4. If both are north or south, and both east or west, add the less course, and the supplement of the greater.

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In each case, the result will give the angle contained by the two sides.

To protract a field according to the preceding rules is preferable to the method of doing it by parallel lines, though it may not be so easy to the learner at first. It is difficult to draw parallel lines with perfect accuracy, and a small deviation from a true line may make considerable difference in the plot of a field.