Note 1.-In the proportion for finding the correction of the latitude or departure, the decimal parts of the sum of the distances and of the particular distance may be omitted, taking, in each case, the nearest number of whole chains. 2. The corrections may be frequently estimated with sufficient accuracy without the trouble of working out the proportions. 3. When one or two of the sides are hilly, or when there are other difficulties in the way of obtaining their bearing or distances with accuracy, it is better to allow a considerable part of the errors, on the latitudes and departures corresponding to them, and afterwards to apportion the remaining part among the others. EXAMPLES. 1. Given the bearings and distances of the sides of a tract of land as follow: 1st. S. 40° E. 31.80 ch.; 2nd. N. 54° E. 2.08 ch.; 3rd. N. 294° E. 2.21 ch.; 4th. N. 281 E. 35.35 ch.; 5th. N. 57° W. 21.10 ch.; 6th. S. 47° W. 31.30 ch. Required the corrected differences of latitude and departures. 2. Given the bearings and distances of the sides of a tract of land as follow: 1st. N. 75° E. 13.70 ch.; 2d. N. 20 E. 10.30 ch.; 3d. East 16.20 ch.; 4th. S. 331° W. 35.30 ch.; 5th. S. 76 W. 16 ch.; 6th. North 9 ch. ; 7th. S. 84° W. 11.60 ch.; 8th. N. 53° W. 11.60 ch.; 9th. N. 36° E. 19.36 ch.; 10th. N. 22° E. 14 ch.; 11th. S. 761° E. 12 ch.; 12th. S. 15° W. 10.85 ch.; 13th. S. 18° W. 10.62 ch.; to the place of beginning. Required the corrected latitudes and departures. Ans. 1st. 3.56 N. 13.26 E.; 2d. 9.66 N. 3.62 E.; 3d. 0.02 N. 16.22 E.; 4th. 29.39. S. 19.44 W.; 5th. 3.85 S. 15.50 W.; 6th. 9.01 N. 0.01 E.; 7th. 1.19 S. 11.52 W.; 8th. 6.96 N. 9.27 W.; 9th. 15.54 N. 11.61 E.; 10th. 12.95 N. 5.38 E.; 11th. 2.73 S. 11.70 E.; 12th. 10.46 S. 2.80 W.; 13th, 10.08 S. 3.27 W. * When, as in this case, the correction is found to be nearly midway between two numbers, it is best to note them both. Then, if in using the one that is nearest to the true value, the sum of the corrections does not equa! the whole error, the other should be taken. CHAPTER II. On supplying omissions in the dimensions of a survey. When the bearings and distances of all the sides of a survey are known, except one bearing and one distance, or two bearings, or two distances, these can be obtained by calculation, provided those that are known can be depended on, as sufficiently accurate. This may sometimes be necessary when there are obstacles in the way of obtaining one or two of the bearings or distances; or when, after they have all been taken on the ground, the notes of one or two of them happen to be obliterated. As, however a bearing, or distance thus obtained, must be affected by any error or errors that may have been made in taking the others, it is better, when practicable, to have the bearings and distances of all the sides, as taken on the ground. PROBLEM I. The bearings and distances of all the sides of a tract of land, except the bearing and distance of one side, being given, to find these. RULE. Find by prob. 11, of the preceding chapter, the differences of latitude and the departures for the sides whose bearings and distances are given, and place them in their proper columns in a table ruled for the purpose: ference of their sums, place it opposite the unknown side, in the column whose sum is the least. The sums of the two columns will then be equal. This is called balancing the latitudes. Do the same with the eastings and westings. The two numbers inserted to make the latitudes and the departures balance, will be the difference of latitude and the departure of the unknown side; with which its bearing and distance may be found, by prob. 10, of the preceding chapter. Note 1.-By the application of this rule, the bearing and distance of a line joining two corners or stations, may be found, when there are obstacles in the way which prevent our going directly from one corner to the other, or when one cannot be seen from the other. To do this, let one or two, or more stations, if necessary, be taken out of the line, and take the bearing and distance from the first corner to the first assumed station; from this station to the second; and so on, to the second corner. Then considering these bearings and distances, as the bearings and distances of the sides of a survey, the required bearing and distance of the line may be found by the above rule. The bearing thus found must be reversed, in order to have the bearing from the first corner to the sccond 2. In the same way the bearing and distance of a straight road to run between two given places, may be found, by taking the several bearings and distances of the old road if there is one; or of lines joining assumed stations and extending from one of the places to the other. EXAMPLES. 1. The bearings and distances of the side of a tract of |