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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.

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

Nors. 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, to the supplement of the greater. 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.*

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TO DRAW A PLOT OF THIS FIELD, ACCORDING TO THE PRECEDING RULES.

Having drawn the side AB, according to the directions before given for laying off the first course and distance, compare the first and second courses together, and they will be found to be both north and both east; consequently, the angle between them is found by RULE 4, as follows: 16° 30/ the less course, and 98° the supplement of the greater being added, the sum is 114° 30', for the angle at B. Compare the second and third courses, and they will be found to be one north and one south, and both east; consequently, according to RULE 2, 82° the second course added to 170 the third course, the sum 99° is the angle at C. The third and fourth courses are both south, and one east and the other west. The angle between them at D is 126°; for 17° the third course added to 37° the fourth course is 54o, which subtracted from 180o leaves 126o, according to RULE 3. The fourth and fifth courses are one south and the other north, and both west. According to RULE 2, 270 the fourth course added to 49o the fifth course, the sum 860 is the angle at E.

A little practice will render this mode of protracting a field familiar and easy, and an attention to the courses will show in what direction the angle is to be made.

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The above field may be protracted, and its area calculated according to the directions given in the preceding EXAMPLES.

SEVERAL FIELD BOOKS TO EXERCISE THE LEARNER IN PLOTTING FIELDS AND CALCULATING THEIR AREA.

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