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Containing three different rules for finding the areas of right-lined figures generally when the bearings and distances of the boundaries are given.
1. Meridians are north and south lines, which are sup-. posed to pass through every station of the
2. The difference of latitude, or the northing or southing of any stationary line, is the distance that one end of the line is north or south from the other end; or it is the distance which is intercepted on the meridian, between the beginning of the stationary line and a line drawn from the other end, perpendicular to that meridian. Thus, if NS, Fig. 77, be a meridian passing through the point A of the line AB, then is Ab the difference of lati tude, or southing of that line.
3. The departure of any stationary line, is the nearest distance from one end of the line to a meridian passing through the other end. Thus Bb, Fig. 77, is the departure or easting of the line AB. But if ns be a meridian, and the measure of the stationary line be taken from B to A, then is BC the difference of latitude, or northing, and AC the departure or westing of the line AB.
4. The meridian distance of any station, is the distance thereof from a meridian passing through the first, or some other particular station of the survey.
5. The Traverse Table is a table containing the difference of latitude and departure corresponding to different courses and distances.
To find the difference of latitude and departure corres-. ponding to any given course and distance, by means of the annexed Traverse Table.
When the distance is any number of whole chains or perches not exceeding 100.
Find the given bearing at the top or bottom of the table according as it is less or more than 45°. Then against the given distance, found in the column of distances at the side of the table, and under the bearing, if at the top, or over it if at the bottom, is the corresponding difference of latitude and departure. The difference of latitude and departure must be taken as marked at the top of the table when the bearing is at the top, but as marked at the bottom, when the bearing is at the bottom. Thus if the bearing and distance be S. 35° 15'E. dist. 79Ch., the diff. of lat. will be 64.51Ch. S. and the dep. 45.59Ch. E.: but if the bearing and distance be S. 54° 45'E. dist. 79Ch. the diff. of lat. will be 45.59Ch. S. and the dep. 64.51.Ch. E.
When the distance is expressed by any whole number of chains or perches exceeding 100.
Divide the given distance into parts that shall not exceed 100 each, and find as before the difference of latitude and departure corresponding to the given bearing and to each of those parts; the sums of the latitudes* and departures thus found will be the latitude and departure required.
* For the sake of conciseness in the expression the word latitude only is
1. A line bears N. 201°E. dist. 117Ch. required the corresponding latitude and departure.
Dist. 100 corresp. Lat. 93.67 and Dep. 35.02
Whole Dist. 117
Lat. 109.59N. Dep. 40.97E.
2. What is the difference of latitude and departure of a line bearing N. 78°W. dist. 243 perches?
Dist. 100 corresp. Lat. 20.36 and Dep 97.90
When the distance is expressed by chains or perches and decimals of a chain or perch.
Find as above the latitude and departure corresponding to the given bearing and to the whole chains or perches Then considering the decimals as a whole number, find the latitude and departure corresponding to it, removing the decimal point in each, two figures to the left hand if there be two decimals, or one figure to the left if there be but one decimal; these added to the former will give the difference of latitude and departure required.
1. If a line bear S. 41°W. dist. 57.36Ch. what will be the corresponding difference of latitude and departure?
Dist. 57.00 corresp. Lat. 42.53 and Dep.37.96.
Whole Dist. 57.36
Lat. 42.80S. Dep. 38.20W.
2. Required the latitude and departure corresponding to a line which bears N. 72°W. dist. 124.37 perches.
Dist. 100.00 cor. Lat. 30.90 and Dep. 95.11
Whole Dist. 124.37
Lat. 38.43N. Dep. 118,29W.
Note. If the number of whole chains or perches be less than 10, and there be but one decimal figure, the latitude and departure may be taken out at one view, by considering the mixed number as a whole one, and taking out the latitude and departure corresponding to it and the given bearing, and removing the decimal point in each, one figure to the left hand. Thus if a line bear N. 231°W. dist. 9.3ch. its difference of latitude will be 8.53ch. N. and its departure 3.71.ch. W.
The bearings and distances of a survey being given, to find the area without the necessity of first protracting
1. Rule a table as in the annexed examples: In the first vertical column on the left hand, place the numbers that designate the stations, in the second the bearings,
2. Find by the traverse table the latitudes and departures answering to the several courses and distances, which place in the four succeeding columns, under N. or S., E or W., according as they are north or south, east or west. Add up the northings and southings, and if the sums are not equal, find their difference, which will be the error of the survey in difference of latitude, which call by the same name as the least sum. Proceed in the same manner with the eastings and westings, and find the error of the departures.
3. Divide each of these errors by the sum of the distances in the third column, extending the division to four decimal places.
4. Multiply the first distance, omitting the decimals, if any, by each of these quotients, and the products will be the corrections in difference of latitude and departure, depending on the first course and distance. Take the first two decimals in each of these corrections, increasing the second by a unit if the third exceed 5, and place them in the 8th and 9th columns according to their name, and in the same horizontal column with the first course and distance. Proceed in the same manner to obtain the other corrections. If the sums of these errors are not equal to the errors in difference of latitude and departure respectively, which in consequence of the decimals neglected will sometimes be the case, alter some of them by a unit in the second decimal to make them so.
5. Apply these corrections to their corresponding dif ferences of latitude and departure, by adding when of the same name and substracting when of different names, and the corrected differences of latitude and departure will be obtained, which place in the four succeeding columns.