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 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 900, 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..

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: 900 added to 16° 30' the less Course, and 80 the Complement of the greater, the sum is 1140 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, 820 the second Course added to 170 the third Course, the sum 990 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 170 the third Course added to 370 the fourth Course is 540, which subtracted from 180° leaves 1260, 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.

K

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.

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.

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.