Let AD represent any course, and NAD=ACB, expressed in degrees and minutes, be its bearing. unit of measure of the the radius of the table (Bk. I., Sec. III., Art. 14). Let AC be the course, and also of natural sines Draw DE and CB parallel to NS, and AE perpendicular to AS. Then will DE be the latitude, and AE the departure of the course, and CB the cosine, and AB the sine of the bearing. From similar triangles we have these proportions, 1 : cos of the bearing :: course : latitude, :: 1 : sin of the bearing :: course : departure. Whence, lat. = course X cos of the bearing, dep. = course X sin of the bearing. We have then the following practical rule for computing the latitude and departure of any course. Look in a table of natural sines for the cosine and sine of the bearing. Multiply each by the length of the course, and the first product will be the latitude, and the second will be the departure of the given course. 1. The bearing is 65° 39', the course 69.41 chains: what is the latitude, and what the departure? Product, which is the Dif. of Latitude, 28.6184371. 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. 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 N H B G F A E 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. 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. Latitude for 55 55. 3602.00 Departure for 7800 . 6919.00 25.40 Departure for 55 48.79 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 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 4730, and the course 37.57. Latitude for 37.00 24.88 Departure for 37.00 .38 Departure for .57 Latitude for .57 27.39 .42 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 southings thus corrected are entered in columns on the right, under the head balanced. The eastings and westings are balanced in the same manner; the difference between their sums being called error in departure. For an example, we will resume the one already con 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 |