G Place the compass at A and take the bearing to B, which is PAB: suppose this angle has been found to be 31. The bearing from A to B is then N 31° W. Enter this B bearing in the field notes opposite station 1. Then measure the distance from A to B, which we will suppose to be 10 ch, and insert that distance opposite station 1, in the 11 column of distances. W We next take the bearing from B to C, N 623o E, and then measure the distance BC=9 ch 25 1, both of which we insert in the notes opposite station 2. At station C we take the bearing to D, S 36° E, and then measure the distance CD=7 ch 601, and place them in the notes opposite station 3. At D we take the bearing to A, S 451° W, and then measure the distance DA=10 ch 40 l. We have thus made all the measurements on the field which are necessary to determine the content of the ground. 134. REMARK I. The reverse bearing, or back sight, from B to A, is the angle ABH; and since the meridians NS and HG are parallel, this angle is equal to the bearing NAB. The reverse bearing is, therefore, S 311° E. The reverse bearing from C, is S 633° W: that is, it is the And generally, a reverse bearing, or back sight, is always equal to the forward bearing, and differs from it in both of the letters by which it is designated. 135. REMARK II. In taking the bearings with the compass, there are two sources of error. 1st. The inaccuracy of the observations: 2d. Local attractions, or the derangement which the needle experiences when brought into the vicinity of iron-ore beds, or any ferruginous substances. To guard against these sources of error, the reverse bearing should be taken at every station: if this and the forward bearing are of the same value, the work is probably right; but if they differ considerably, they should both be taken again. 136. REMARK III. In passing over the course AB, the northing is found to be HB, and the departure, which is west, is represented by AH. Of the course BC, the northing is expressed by BG, and the departure, which is east, by GC. Of the course CD, the southing is expressed by CI, and the departure, which is east, by CF. Of the course DA, the southing is expressed by KA, and the departure, which is west, by DK. It is seen from the figure, that the sum of the northings is equal to HB+BG=HG; and that the sum of the southings is equal to CI+KA=PA=HG : hence, the sum of the northings is equal to the sum of the southings. If we consider the departures, it is apparent that the sum of the eastings is equal to GC+CF=GF; and that the sum of the westings is equal to AH+DK=GF: hence also, the sum of the eastings is equal to the sum of the westings. We therefore conclude, that when any survey is correctly made, the sum of the northings will be equal to the sum of the southings, and the sum of the eastings to the sum of the westings. It would indeed appear plain, even without a rigorous demonstration, that after having gone entirely round a piece of land, the distance passed over in the direction due north, must be equal to that passed over in the direction due south; and the distance passed over in the direction due east, equal to that passed over in the direction due west. Having now explained the necessary operations on the field, we shall proceed to show the manner of computing the content of the ground. We shall first explain THE TRAVERSE TABLE. 137. This table shows the difference of latitude, and the departure, corresponding to any bearing, and for courses less than 100. Let AB denote any course, NS the meridian, and NAB the bearing of AB. Then will AC be the difference of latitude, and BC the departure. W N E A It is evident that the course, the difference of latitude, and the departure, are respectively, the hypothenuse, the base, and the perpendicular of a right-angled triangle, of which the bearing is the angle at the base. S If there be two bearings, which are complements of each other, or of which the sum is 90°, the difference of latitude corresponding to the one, will be the departure of the other, and reciprocally. For, if BC were a meridian, CBA which is the complement of CAB, would be the bearing of BA; CB would be the difference of latitude, and C. would be the departure. In the traverse table, the figures at the top and bottom of each page, show the bearings to degrees and parts of a degree; and the columns on the left and right, the distances to which the latitudes and departures correspond. If the bearing is less than 45°, the angle will be found at the top of the page; if greater, at the bottom. Then, if the distance is less than 50, it will be found in the column "distance," on the left hand page; if greater than 50, in the corresponding column of the right hand page. The table is calculated only to quarter degrees, for the bearings cannot be relied on to smaller parts of a degree. The latitudes or departures of courses of different lengths, but which have the same bearing, will be proportional to the lengths of the courses. Thus, in the last figure, the latitudes AG, AC, or the departures GF, CB, are to each other as the courses AF, AB. Therefore, when the distance is greater than 100, it may be divided by any number which will give an exact quo found and multiplied by the divisor, the products will be the latitude and departure of the whole course. It is also plain, that the latitude or departure of two or more courses, having the same bearing, is equal to the sum of the latitudes or departures of the courses taken separately. Hence, if we have any number greater than 100, as 614, we have only to regard the last figure as a cipher, and recollect that, 610+4=614; and also, that the latitude and departure of 610, are ten times greater, respectively, than the latitude and departure of 61: that is, equal to the latitude and departure of 61 multiplied by 10, or with the decimal point removed one place to the right. 1. To find the latitude and departure for the bearing 291, and the course 614. In this example, the latitude and departure answering to the bearing 2910, and to the distance 61, are first taken from the table, and the decimal point removed one place to the right this gives the latitude and departure for the distance 610; the latitude and departure answering to the same bearing and the distance 4, are then taken from the table and added. 2. To find the latitude and departure for the bearing 6210, and the course 7855 chains. Latitude for 7800 . 3602.00 Departure for 7800 55. 25.40 Departure for Latitude for 3627.40 Departure for 7855 6919.00 . 48.79 6967.79 REMARK. When the distances are expressed in whole numbers and decimals, the manner of finding the latitudes and departures is still, the same, except in pointing off the places for decimals: but this is not difficult, when it is remembered that the column of distances in the table, may ve regarded as decimals, by removing the decimal point to the left in the other columns. 3. To find the latitude and departure for the bearing 4730, and the course 37.57. 138. The use of the traverse table being explained, we can proceed to compute the area of the ground. The field notes having been completed, rule a new table, as below, with four additional columns, two for latitude, and two for departure. Then find, from the traverse table, the latitude and departure of each course, and enter them in the proper columns opposite the station. This error takes the For example, if sum of the south Then add up the column of northings, and also the column of southings: the two sums should be equal to each other. If they are not, subtract the less from the greater, and the remainder is called the error in latitude. name of that column which is the least. the sum of the northings is less than the ings, the error is called, error in northing: but if the sum of the southings is less than the sum of the northings, the erro is called, error in southing. We find the error for each par tieular course by the following proportion. As the sum of the courses Is to the error of latitude, So is each particular course To its correction. The error of each course, thus found, may be entered in a separate column; after which, add it to the latitude of the course, when the error and latitude are of the same name, but subtract it, when they are of different names. This will make the sum of the northings equal to the sum of the southings, and is called balancing the work. The northings and southings, thus corrected, are entered in columns on the right, under the head, balanced. Having done this, balance the eastings and westings in the very same manner. The dif |