When the field is very irregularly shaped, it will often happen that parts of the same area will be included in two or more products in the column of areas, but as often as a space is included in a north product, so often will it be included in a south product, so that the difference between the sum of N. Areas and the sum of S. Areas will, in every case, give the area included by the boundary lines of the field. 117. The following summary will serve as a rule for finding the area of any field bounded by any number of straight lines: RULE. 1. Prepare a table with columns marked as in the example, viz.: No., Courses, Rods (or Chains, as the case may be), N., S., E., W., D. M. D., M. D., N. Areas, S. Areas. 2. Beginning at the most westerly point, write the courses in the table. Find from the Traverse Table the northings, southings, eastings, and westings of the several sides of the field, and write them opposite their respective courses under their proper letters N., S., E., or W. 3. Add the numbers in the columns marked N., S., E., and W., and balance the work if necessary. 4. To form the column D. M. D., write the first easting on the first line, then add it to itself, and write the sum immediately beneath; add to this the next easting, or, if a westing, subtract, —and write the result on the second line, and so continue adding each easting twice, except the first, and subtracting each westing twice, except the last. 5. To form the column M. D., divide by two the several numbers standing ON THE LINE in the column D. M. D. 6. Multiply each number in the column M. D. by the northing, or southing, on the same line, and write the product in the column of N. Areas or S. Areas, according as the multiplier is in the column N. or S. 7. Subtract the sum of N. Areas from the sum of S. Areas, and the remainder will be the area of the field in units of square measure, corresponding to the linear unit in the column of dis tances. 118. Ex. Required, the area of a tract of land of which the following are the field notes: 119. In the surveys already given, the field-book is arranged as though the surveyor began at the most westerly point of the field, and went round so as to keep the field on his right. In calculating areas of fields, it is better to begin, in all cases, at the same point, i. e. at the most easterly or westerly point of the field. Surveyors, and those who have written upon this subject, all agree in beginning at the most westerly point, and going round so as to have the field on the right. If the surveyor traverses the lines so as to have the field on the left, the courses must all be reversed, otherwise the S. Areas will represent the part without the field, and the N. Areas the whole within and without. When there is any difficulty in deciding, by inspecting the field-book, which is the most westerly point, plot the field and the required point will become evident. TO PLOT A FIELD. 120. This can be done with only the dividers and rule described in Chapter III. Place the rule anywhere on the 620 5/910 ing the degrees; once placed, the centre of that edge of the rule which contains the centre from which the degrees were marked is the place of beginning, and this edge of the rule is the meridian passing through the place of beginning. With a sharp point mark the place upon the paper required by the number of degrees mentioned in the first course, counting, if the course is north, from the top round toward the right, if the course is south, from the bottom toward the right. This is upon the supposition that the plotting begins at the most westerly corner of the field. With a sharp point draw an indefinite line from the place of beginning (mentioned above) through the place just marked, and then, from the scale of equal parts, lay off from the place of beginning, on the indefinite line, the given length. Having marked the termination of this line with the sharp point, and, to avoid any mistake, enclosed it with a small circle with pen or pencil, place the rule with its centre at this termination, and with the line itself under the same number upon the rule as before. This brings the rule into a position parallel to its first position. Then mark off the degrees in the second course, and proceed the same as with the first line, and so continue round the field. When the course changes to the west, place the rule parallel to its first position, in the same manner as before, only having the edge on which the degrees are marked face the other way, i. e. having the rule on the other side of the meridian. Suppose, for example, it be required to protract the field from the field-book in Art. 118. The several positions of the rule are shown in the figure on page 83. 121. When the survey is correct, and the plot carefully. drawn, the end of the last course will fall on the place of beginning. After the plan has been made, copies may be taken by placing under this plan as many sheets of paper as the number of plans required, pressing a sharp point through the several corners of the field and the paper beneath, and then joining the points made on each sheet in the same manner as they are in the first plan. Upon the plan should be written the owner and the area; the courses, each written in the direction specified by the course, upon their respective sides; the names of persons owning around the field, in their proper places; the town, and part of the town, in which the field is; the name of the surveyor, and the date of the survey; on the side of the paper which is the north, the letter "N"; and any other data which may be useful to |