(156.) When a field has been correctly surveyed, and the latitudes and departures accurately calculated, the sum of the northings should be equal to the sum of the southings, and the sum of the eastings equal to the sum of the westings. If the northings do not agree with the southings, and the eastings with the westings, there must be an error either in the survey or in the calculation. In the preceding example, the northings exceed the southings by one link, and the westings exceed the eastings by five links. Small errors of this kind are unavoidable; but when the error does not exceed one link to a distance of three or four chains, it is customary to distribute the error among the sides by the following proportion : As the perimeter of the field, Is to the length of one of the sides, So is the error in latitude or departure, To the correction corresponding to that side. This correction, when applied to a column in which the sum of the numbers is too small, is to be added; but if the sum of the numbers is too great, it is to be subtracted. We thus obtain the corrections in columns 8 and 9 of the preceding table; and applying these corrections, we obtain the balanced latitudes and departures, in which the sums of the northings and southings are equal, and also those of the eastings and westings. As the computations are generally carried to but two decimal places, the corrections of the latitudes and departures are only required to the nearest link, and these corrections may often be found by mere inspection without stating a formal proportion. Thus, in the preceding example, since the departures require a correction of five links, and the field has five sides which are not very unequal, it is obvious that we must make a correction of one link on each side. It is the opinion of some surveyors that when the error in latitude or departure exceeds one link for every five chains of the perimeter, the field should be resurveyed; but most surveyors do not attain to this degree of accuracy. The error, however, should never exceed one link to a distance of two or three chains. (157.) To find the area of the field. Let ABCDE be the field to be measured. Through A, the most western station, draw the meridian NS, and upon it let fall the perpendiculars BF, CG, DH, EI. Then the area of the required field is equal to FBCDEI—(ABF+AEI). But FBCDEI is equal to the sum of the three trapezoids FBCG, GCDH, HDEI. Also, if the sum of the F G A H I N S E B D C parallel sides FB, GC be multiplied by FG, it will give twice the area of FBCG (Art. 87). The sum of the sides GC, DH, multiplied by GH, gives twice the area of GCDH; and the sum of HD, IE, multiplied by HI, gives twice the area of HDEI. Now BF is the departure of the first side, GC is the sum of the departures of the first and second sides, HD is the algebraic sum of the three preceding departures, IE is the algebraic sum of the four preceding departures. Then the sum of the parallel sides of the trapezoids is obtained by adding together the preceding meridian distances two by two; and if these sums are multiplied by FG, GH, &c., which are the corresponding latitudes, it will give the double areas of the trapezoids. (158.) It is most convenient to reduce all these operations to a tabular form, according to the following RULE. Having arranged the balanced latitudes and departures in their appropriate columns, draw a meridian through the most eastern or western station of the survey, and, calling this the first station, form a column of double meridian distances. The double meridian distance of the first side is equal to its departure; and the double meridian distance of any side is equal to the double meridian distance of the preceding side, plus its departure, plus the departure of the side itself. . Multiply each double meridian distance by its corresponding northing or southing, and place the product in the column of north or south areas. The difference between the sum of the north areas and the sum of the south areas will be double the area of the field. It must be borne in mind that by the term plus in this rule is to be understood the algebraic sum. Hence, when the double meridian distance and the departure are both east or both west, they must be added together; but if one be east and the other west, the one must be subtracted from the other. The double meridian distance of the last side should always be equal to the departure for that side. This coincidence affords a check against any mistake in forming the column of double meridian distances. The preceding example will then be completed as follows: Twice the figure FBCDEI is 372.2098 square chains. The difference is 289.9178 66 Therefore the area of the field is 144.9589 square chains, or 14.49589 acres, which is equal to 14 acres, 1 rood, 39 perches. Ex. 2. It is required to find the contents of a tract of land of which the following are the field notes: |