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. Ch. L. O E. 19.60 EA. N. 49 0 W. 25.20 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 8° 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 Easterły ; consequently, according to Rule 2. 82° the second Course added to 17° 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 126° ; for 17° the third Course added to 37° the fourth Course is 54°, which subtracted from 180° 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. 37° the fourth Course added to 49° the fifth Course, the Sum 86° 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. Ch. L. 0 E. 8.47 LM. S. 63 30 W, 13.44 Acres Rood Rods Area 167 el 30 The above Field may be protracted, and its Area calculated according to the directions given in the preceding EXAMPLES A Rule to determine whether the Courses in any Survey have been accurately taken. By the Rules for protracting a Field, Page 48, find the Quantity of the several Angles, and add the whole together; to their Sum add 360°; divide this Sum by 180°; and, if the Survey is right, the Quotient will equal the number of Angles contained in the Field. Thus, in the preceding Example, the Sum of all the Angles is 1980°; to this add 360° and it makes 2340°; this Sum being divided by 180° the Quotient will be 13, which is the number of Angles in the field. See the Figure. When the Angle is without the Field, as at B, F, G and H, subtract the Quantity of the Angle, as found by the preceding directions, from 360 and make use of the Remainder in adding the several Angles. Thus the Angle at B 150° 15' must be subtracted from 360°, and the Remainder 209° 45' considered as the real Quantity of that Angle. If there is an error, the Field must be re-surveyed, and the error corrected, else the true Area cannot be ascertained. Note. Directions will be given in SECTION III. for determining whether the sides have been accurately measured. Fig. 54. Demonstration of the preceding Rule. Suppose a Plot of a Fild, as ABCD, &c. PLATE II. From some point within the field, as at a, draw Lines to the several Angles; and it is evident the whole will be divided into as many Triangles as there are Sides to the Field, that is 7. Now, as the three the Angles of all these Triangles will be 7 times 180°, that is 1260°. The Sum of the Angles at the Centre is 360°, because the Arches which measure those Angles form a Circle. Therefore, 360° the Sum of those central Angles, subtracted from 1260° will leave the Sum of all the other Angles ; which are the Angles made by the several Sides of the Field. The Angles of this field will be found to contain 900°; if to this you add 360° and divide the Sum, viz. 1260° by 180° the Quotient will be 7, the number of the Sides or Angles of the Field. Several Field Books to exercise the Learner in plotting Fields and calculating their Area. NO. V. Rods. Rood Rods 1 34 12. N. 87 O E. 29.92 2.23 2.50 11.75 Acres Roods Rods NO. VI. Rods 69.4 O E. 4. 12.5. 14. Acres Rood Rods 1 O E. NO. VIII. Rods. O E. 10.68 0 E. 13.3 9.8 Acres Rood Rods 12 NO. VII. Ch. L. |