2. The bearing is 75° 47', the course 89.75 chains: what is the latitude, and what the departure? Product, which is the Dif. of Latitude, 22.0417025. Natural sine of 75° 47′ Length of course Product, which is the Departure . .96937 87.0009575. 20. In this manner the traverse table given at the end of the book has been computed. When the bearing is given in degrees and quarters of a degree, and the difference of latitude and departure are required to only two places of decimals, they may be taken directly from the traverse table. 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 latitudes or departures of courses of different lengths, but which have the same bearing, are proportional to the lengths of the courses. Thus, in the W figure, the latitudes AG, AC, or the departures GF, CB, are to each other as the courses AF, AB. N H G E A S Therefore, when the distance is greater than 100, it may be divided by any number which will give an exact quotient, less than 100: then the latitude and departure of the quotient being 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 recollect that, 610+4= 614; and as great, respectively, as 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. EXAMPLES. 1. To find the latitude and departure for the bearing 2910, and the course 614. Latitude for 610 . . 530.90 Departure for 610 . . 300.40 Latitude for 3.48 Departure for 4. 4. 1.97 Latitude for 614. . 534.38 Departure for 614., 302.37 In this example, the latitude and departure answering to the bearing 2940, 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. 55. 48.79 Latitude for 7800. 3602.00 Departure for 7800. 6919.00 25.40 Departure for 55 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 be 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 473, and the course 37.57. Latitude for 37.00 24.88 Departure for 37.00 . 27.39 Latitude for .57 .57 .42 .38 Departure for OF BALANCING THE WORK. 21. 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. 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; the remainder is called the error in latitude. This error takes the name of that column which is the less. For example, if the sum of the northings is less than the sum of the southings, the error is called, error in northing: but if the sum of the southings is less than the sum of the northings, the error is called, error in southing. We find the error for each particular course by the following proportion. As the sum of the courses The error 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 south ings thus corrected are entered in columns on the right, under the head balanced. The eastings and westings are balanced in the same the difference between their sums being called error in departure. manner; For an example, we will resume the one already con As 37.25 0.06 :: 10 : 0.02* error in dep. of 1st course. As 37.25 0.06 :: 9.25 0.01 error in dep. of 2d course. As 37.25 0.06 :: 7.60 0.01 As 37.25 0.06: 10.40 : 0.02 error in dep. of 3d course. error in dep. of 4th course. 22. REMARK I. In finding the error in latitude or departure, for a particular course, the last figure is sometimes. doubtful in which case it is best to mark it, as in the third proportion for error in latitude, and the first for error in departure; and then, if the figures taken do not balance the work, let each be increased or diminished by 1. 23. REMARK II. It has already been observed (Art. 18), that if the measurements on the field be correctly made, the sums of the northings and southings will be equal to each other, as also those of the eastings and westings. It is the opinion of some surveyors, that when the error in latitude or departure exceeds one link for every five chains of the courses, the field notes ought not to be relied on. This, perhaps, is a higher degree of accuracy than can be attained. The error, however, should always be made. considerably less than one link to a chain. 24. The following is an example in which the latitude and departure of each course have been computed from Instead of balancing by the method just explained, we divide each error by two. Now if we subtract half the error in southing from the column of northings, and at the same time add it to the column of southings, these two columns will exactly balance. In like manner, if we subtract half the error in easting from the column of westings, and at the same time add it to the column of eastings, these columns will also balance. The errors should be distributed in proportion to the lengths of the courses, but this may be done with sufficient accuracy without making the proportions. If any of the courses have been run over rough ground, the probability is that the errors belong to these courses, and they should be distributed among them. Lee to In this example we separate the half error in southing into the three parts .40700, .21302, and .04674, and subtract them respectively from the northings of courses 2, 1, and 3, and then place the northings in the balanced columns. For the southings we separate the half error into the four parts .40772, .20031, .03121, and .02752, and add them respectively to the southings of the courses 4, 5, 8, and 7. We then enter the southings in the balanced columns. As the error in easting is so small, we add half of it to the easting of course 3, and subtract half from the westing of |