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The dimensions of the buildings, their distances apart, and the direction of one side of each being taken, sufficient data is afforded for locating them, correctly, upon the map.
NOTE.—The advantage of the compass over other instruments with which angles are measured, lies chiefly in this: that the Bearing of a course may be measured at any point on the line.
When the angle between adjacent sides is taken with the Transit or Theodolite, the work can only be done at the corners of the field; and when, as frequently happens, a hill intervenes between two consecutive stations, it becomes necessary to locate a point on the hill, in the true line, and then return to the corner to measure the angle; whereas, when the compass is employed, the establishment of the intermediate point on the hill affords the means of taking the proper bearing without going to the angle. Furthermore, the bearings may be measured with the compass, by placing it at the alternate stations only.
CONTENTS Of Ground.
84. Having explained the necessary operations on the field, we shall now proceed to show the manner of computing the contents of ground.
THE TRAVERSE TABLE AND ITS USES.
85. This table shows the latitude and departure corresponding to bearings that are expressed in degrees and quarters. of a degree, from 0 to 90°, and for every course from 1 to 100, computed to two places of decimals.
The following is the method of deducing the formulas for computing a traverse table; by means of these formulas and a
table of natural sines, the latitude and departure of a course may be computed to any desirable degree of accuracy.
Let AD represent any course, and
NAD ACB, expressed in degrees and
minutes, be its bearing. Let AC be the unit of measure of the course, and also the radius of the table of natural sines (Bk. I., Sec. III., Art. 57). 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, to the radius AC 1.
From similar triangles we have these proportions,
1 : cos of the bearing :: course : latitude;
: sin of the bearing :: course : departure.
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?
2. The bearing is 75° 47', the course 89.75 chains: what
is the latitude, and what the departure?
Product, which is the Departure
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.
When the bearing is less than 45°, the angle will be found at the top of the page; when greater, at the bottom. When the distance is less than 50, it will be found in the column 'distance," on the left-hand page; when greater than 50, in the corresponding column of the right-hand page.
86. The latitudes or departures of courses of different lengths, but which have the same bearing, are proportional to to the lengths of the courses. Thus, in the figure, the latitudes AG, AC, or the departures GF, CB, are to each other as the
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, 6104614; and also, that the latitude and departure of 610, are ten times as great, respectively, as the latitude and departure of 61: that is, equal to the latitude and departure of 61 multiplied by 10; or, to such latitude and departure with the decimal point removed one place to the right.
1. To find the latitude and departure for the bearing 2940,
In this example, the latitude and departure answering to the bearing 294°, 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 62°, and the course 7855 chains.
NOTE. 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 simply 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.
87. Having explained the use of the traverse table, we can proceed to compute the area of the ground.
The field-notes having been completed, rule a new table, as on next page, 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 least. 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