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 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 calculations, 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 Northerly and the other Southerly, one Easterly and the other Westerly, subtract the less Course from the greater, the Remainder will be the Angle between them.

RULE 2. If one is Northerly and the other Southerly, and both Easterly or Westerly, add both Courses together; the Sum will be the Angle between them.

RULE 3. If both are Northerly or Southerly, and one Easterly and the other Westerly, subtract the Sum of both from 180°; the Remainder will be the Angle between them.

RULE 4. If both are Northerly or Southerly, and both Easterly or Westerly, add 90°, the less Course and the Complement of the greater together; the Sum will be the Angle between them.

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, particularly without a parallel Rule; and a small deviation from a true Line may make considerable difference in the Plot of a Field.

EXAMPLE II.

FIELD BOOK. See PLATE III. Fig. 58.

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 Northerly and both Easterly; consequently the Angle between them is found by RULE 4. as follows: 90° added to 16° 30' the less Course and 8o the Complement of the greater, the Sum is 114° 30' for the Angle at B. Compare the second and third Courses, and they will be found to be one Northerly and one Southerly and both Easterly; consequently, according to RULE 2. 82o the second Course added to 17o the third Course, the Sum 99° is the Angle at C. The third and fourth Courses are both Southerly and one Easterly and the other Westerly. The Angle between them at D is 1260; for 17° the third Course added to 370 the fourth Course is 54° which subtracted from 180o leaves 126°, according to RULE 3. The fourth and fifth Courses are one Southerly and the other Northerly and both Westerly. According to RULE 2. 370 the fourth Course added to 49° 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.

EXAMPLE III.

FIELD BOOK. See PLATE IV. Fig. 66,

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 calcula

ting their Area.

Rods.

No. I.

Rods.

No. III.

CASE V.

To survey a Field from one Station, at any place within the Field, from which the several Angles may be seen. Take the Bearing of the Angles, and measure their Distance from the Station.

FIELD BOOK. See PLATE III. Fig. 61.

From Station to A. N. 20o W.

Ch. L.

8.70

F. N. 65 W

To protract this Field.

Draw a Meridian Line as N. S. From some point in that Line as a Centre lay off the Bearing and Distance to the several Angles, and draw Lines from one Angle to another, as AB, BC, CD, &c.

To find the Area.

The Area may be calculated according to PROB. XII. by measuring Diagonals and Perpendiculars; or more accurately according to PROB. IX. Rule 4.

As the Bearing and Distance of the Lines from the Station to the several Angles are known, two Sides and their contained Angle are given in each of the Triangles into which the Plot is divided; the Area may, therefore, be readily calculated by the Rule above referred to..

Note. As in the operation, the Logarithm of Radius is to be subtracted from the Sum of the other Logarithms, it may be done by rejecting the Left hand figure, without the trouble of putting down the Cyphers and subtracting. Triangle aAB.

a A, 8. 70

Triangle aCD.

aB, 10

0.93952 aC, 11.40
1.00000 aD, 10.50

1.05690

1.02119

Sine AaB, 80°