2. Given the bearing of a line, S. 32° 30′ E., and the departure 10.96 ch. to find the distance and difference of latitude. Ans. Dist. 20.40 ch.; diff. lat. 17.20 S.

3. Given the distance of a line, running between the north and east, 44 ch. and its difference of latitude 34.43 ch.; to find the bearing and departure.

Ans. Bearing, N. 38° 30′ E.; dep. 27.39 ch. E.

4. The bearing of a line S. 32° 30′ E., and the difference of latitude 17.21 ch. being given, to find the distance and departure. Ans. Dist. 20.41 ch.; dep. 10.96 E.

5. Given the difference of latitude of a line 27.92 N., and the departure 5.32 E.; to find the bearing and disAns. Bearing, N. 10° 47′ E.; dist. 28.42.

tance.

6. The distance of a line, running between the north and west, is 35.35 ch., and its departure 15.08 ch., required the bearing and difference of latitude.

Ans. Bearing N. 25° 15' W.; diff. lat. 31.97 N.

PROBLEM XI.

To find the difference of latitude and departure corresponding to any given bearing and distance, by means

of the

Traverse Table.

RULE.

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,

the given distance, found in the column of distances at the side of the table, and under or over the given bearing, is the difference of latitude and departure; which 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.

When the distance is a number of whole chains or perches, exceeding 100.

Separate the 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 these will be the difference of latitude and departure required.

When the distance is expressed by chains or perches and the decimal of a chain or perch.

Find, as above, the difference of latitude and departure corresponding to the given bearing and to the whole chain or perches. Then considering the decimal part as a whole number, find the difference of latitude and departure corresponding to it, and remove the decimal point in each of them, two figures to the left hand if there are two decimal figures in the distance, or one figure to the left if there is but one; then these added to the former will give the difference of latitude and departure required.

Note. When the number of whole chains or perches is less than 10, and the second decimal figure is a cipher, the difference of latitude and departure may be taken out at once, by considering the mixed number, rejecting

the cipher, as a whole number. The difference of latitude and departure thus found, must have the decimal point in each, removed one figure to the left hand.

EXAMPLES.

1. Given the bearing of a line S. 35° E., dist. 79 ch.; required the difference of latitude and departure by the traverse table. Ans. Diff. lat. 64.51 S.; dep. 45.59 E.

2. A line bears N. 20° E., 117 ch.; required the difference of latitude and the departure.

Dist. 100, gives diff. lat. 93.67 and dep. 35.02

17

15.92

5.95

Whole dist. 117 diff. lat. 109.59 N. dep. 40.97 E.

3. Required the difference of latitude and the departure of a line which bears, S. 41° W., 57.36 ch.

Dist. 57.00 gives diff. lat. 42.53 and dep. 37.96

36

.27

.24

Whole dist. 57.36 diff. lat. 42.80 S. dep. 38.20 W.

4. Required the difference of latitude and departure of a line which bears, N. 72° W., 124.37 ch.

Dist. 100.0 gives diff. lat. 30.90 and dep. 95.11

24.00-
.37-

7.42
.11

22.83

.35

5. Given the bearing and distance of a line, N. 39° W. 15.20 ch., to find its difference of latitude and departure. Ans. Diff. lat. 11.72 N., and dep. 9.67 W.

6. The bearing and distance of a line are N. 46° E., 27.25 ch.; required its difference of latitude and departure. Ans. Diff. lat. 18.93 N. and dep. 19.60 E.

7. The bearing and distance of a line are S. 37° W., 137.50 ch.; required its difference of latitude and departure. Ans. Diff. lat. 109.45 S., and dep. 83.23 W.

8. Required the difference of latitude and departure of a line, whose bearing and distance are S. 61° E., 5.60 ch. Ans. Diff. lat. 5.56 S., and dep. 0.63 E.

PROBLEM XII.

Given the bearings and distances of all the sides of a tract of land to obtain the corrected latitudes and departures.

RULE.

1. Rule a table as in the annexed example, in the first vertical column of which, place the letters designating the sides, or the numbers denoting the stations at the beginning of each side; in the second column, place the bearings; and, in the third, the distances.

2. Find, by the last problem, the difference of latitude and the departure, corresponding to each side, and place them in the next four columns, under their proper heads of N. or S., E. or W. Add up the northings and south

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 in departure. Also add up the column of distances. Then it will be,

As the sum of the distances,

Is to any particular distance,

So is the error in latitude or departure

To the correction of latitude or departure, corresponding to that distance.

3. Find, by the above proportion, the corrections of latitude and departure corresponding to all the sides, calculating them to the nearest two decimal figures, and place them in the next two columns, heading them with the same names as the errors in latitude and departure. If the sums of these corrections, are not respectively equal to the errors in latitude and departure, which, in consequence of the fractions neglected, will sometimes be the case, alter some of them by a unit in the second decimal figure, to make them so.

4. Apply these corrections to their corresponding differences of latitude and departures, by adding when of the same name, but by subtracting when of different names, and the corrected differences of latitude and departures will be obtained; which may be placed in the four succeeding columns.

In these the sums of the northings and southings will be equal, and also those of the eastings and westings.*

* The directions given in the rule, for correcting the errors in difference of latitude and departure, are deduced from the rule given and demonstrated in No. 4, of a periodical work, called the Analyst, by Nathaniel Bowditch, A. M., and also by the editor, Professor Adrain. The demonstration is too