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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. 351° 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. 201° 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. 418° 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. 72o W., 124.37 ch.

Dist. 100.0 gives diff. lat. 30.90 and dep. 95.11
24.00
7.42

22.83
.37-
.11

.35

Whole dist. 124.37 diff. lat. 38.43 N. dep. 118.29 W.

5. Given the bearing and distance of a line, N. 391° 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. 374° 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 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 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, correspond-

ing 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

Note 1.-In the proportion for finding the correction of the latitude or departure, the decimal parts of the sum of the distances and of the particular distance may be omitted, taking, in each case, the nearest number of whole chains.

2. The corrections may be frequently estimated with sufficient accuracy without the trouble of working out the proportions.

3.. When one or two of the sides are hilly, or when there are other difficulties in the way of obtaining their bearing or distances with accuracy, it is better to allow a considerable part of the errors, on the latitudes and departures corresponding to them, and afterwards to apportion the remaining part among the others.

EXAMPLES.

1. Given the bearings and distances of the sides of a tract of land as follow : 1st. S. 40° E. 31.80 ch.; 2nd. N. 54° E. 2.08 ch.; 3rd. N. 29° E. 2.21 ch.; 4th. N. 281° E. 35.35 ch.; 5th. N. 57° W. 21.10 ch.; 6th. S. 47° W. 31.30 ch. Required the corrected differences of latitude and departures.

Sta. Courses.

Dist.

Cor. Cor.
Ch. N. L. S. L. E. D. W.D. s. E. N. L. S. L. E. D.W.D.

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[ocr errors]

| 123.84 15.64 45.52 40.41| 40.58.12.17 145,58 15.58 40.5140,5) 45.52

40.41

12 Er. S.

.17 Er. E.

As 124 : 32 :: .12 : .03

124 : 2 :: .12 : .00

24 : 35 :: .12 : .03 or .04
124 : 21 :: .12 :.02
124 : 31 :: .12 : .03

As 124 : 32 : : .17 : .04 or .05*

124 : 2 ::.17 : .00
124 : 35 ::.17 : .05
124 : 21 ::.17 : .03
124 : 31 ::.17 :.04

2. Given the bearings and distances of the sides of a tract of land as follow : 1st. N. 75o E. 13.70 ch.; 2d. N. 20% E. 10.30 ch.; 3d. East 16.20 ch.; 4th. S. 331° W. 35.30 ch.; 5th. S. 76 W. 16 ch. ; 6th. North 9 ch.; 7th. S. 84° W. 11.60 ch.; 8th. N. 531° W. 11.60 ch.; 9th. N. 36° E. 19.36 ch.; 10th. N. 22° E. 14 ch.; 11th. S. 76° E. 12 ch.; 12th. S. 15° W. 10.85 ch.; 13th. S. 18° W. 10.62 ch.; to the place of beginning. Required the corrected latitudes and departures. Ans. lst. 3.56 N. 13.26 E. ; 2d. 9.66 N. 3.62 E.;

3d. 0.02 N. 16.22 E.; 4th, 29.39.S. 19.44 W.; 5th. 3.85 S. 15.50 W.; 6th. 9.01 N. 0.01 E. ; 7th. 1.19 S. 11.52 W.; 8th. 6.96 N. 9.27 W.; 9th. 15.54 N. 11.61 E. ; 10th. 12.95 N. 5.38 E. ; 11th. 2.73 S, 11.70 E. ; 12th, 10.46 S. 2.80 W.; 13th, 10,08 S. 3.27 W.

* When, as in this case, the correction is found to be nearly midway between two numbers, it is best to note them both. Then, if in using the one that is nearest to the ue value, the sum of the corrections does not equa? the whole error, the other should be taken.

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