Triangles and Trapezoids on the Left. 111000 triangle ACC 213548 trapezoid CCFn 68150 triangle dik 600584 sum 938864 sum brought forward 1539448 sum total Whole area 7a. 2r. 31 p. 3. It is required to lay down a field and find its area, from the PROBLEM VI. Fields comprehended under any Number of Crooked or When a field is bounded by crooked fences, measure a line as near to each as the angles or curves will permit; in doing which, take an offset to each corner or angle in the fence. Where the fences are curved, those offsets must be so taken that a right line drawn from the end of any one perpendicular to the end of the next, on each side, would neither exclude any part of the land to be measured, nor include any of that which is adjacent. Perpendiculars thus erected will divide the whole offset into right-angled triangles and trapezoids, the areas of which are found as before. NOTE 1.-If the curves be so large that many of the offsets would be 2, 3, 4, or 5 chains long, it will be more expeditious and accurate to measure the base without taking any offsets, except such as are short, leaving stations in proper places along the base, to which, when you have obtained its length, you may return, and from them run fresh station-lines to some convenient point, or points, in the curved fence. Upon these lines take offsets as before. (See Example 3.) 2. If any of the fences be curved inward, it is frequently most convenient to measure a line on the outside of the field, and upon it erect perpendiculars to the curved fence, which, in this case, are called insets; and the area thus included must be subtracted from the area of the whole figure. (See Example 4.) 3. When the fences and ditches are to be measured with the field to which they belong, it is generally most practicable to fix the stations within the fences, at a little distance from the corners, and then to measure to the roots of the quickwood, adding or subtracting 5 or 6 links, according to the custom of the place, for the breadth of the ditch. (See Example 5.) 4. When the offsets are small, their places on the base line may be determined by laying the offset staff at right angles upon the chain; but when large, and accuracy is required, they must be found by the cross, and measured by the chain. 5. The base of each triangle and trapezoid, forming an offset, may be found by subtracting the distances on the chain line from each other. 6. The following three methods of finding the area of offsets are erroneous. Some divide the sum of the offsets by their number, for a mean breadth; others divide that sum by one more than their number, for a mean breadth; and both multiply the whole base by the mean breadth, thus supposed to be found, for the area of the whole offset. The first of these methods generally gives the area too much; and the second sometimes too much and sometimes too little. A third method, which is usually more accurate than either of the preceding ones, is to set down each offset twice (accounting that one where the boundary meets the station-line) except the first and last, which are only entered once. The sum of these offsets is then multiplied by the base, the product divided by the number of offsets set down, and the quotient given as the area required. 7. Directions for laying down offsets by a plotting-scale may be seen in Part II. D |