As diff. of lat. of DA, 27.28 S.
: dep. of do. 17.91 W.
:: rad.

Ar. Co. 8.56416

1.25310 10.00000

: tang. of bearing of DA, S. 33° 17' W.

9.81726

59 51=half sum of the angles DAH and AHD.

ADH 60° 17' Ar. Co. 0.06124
As sin. ADHXAD,
AD 32.63

8.48638 : twice ADH 309.95

2.49129 10.00000

:: rad.

As sin. AHD 100° 27'
: sin. ADH 60 17
:: AD 32.63

Ar. Co. 0.00726

9.93876
1.51362

: AH 28.82

1.45964

Hence AH bears N. 52° 32' E. dist. 28.82 ch.

2. Given as follow: 1st side. S. 78° 00' W. dist. 8 ch. 2nd. N. 26° 30' W. dist. 11.08 ch. 3rd. N. 38° 30' E. dist. 12.82 ch. 4th. S. 64° 00' E. dist. 10.86 ch. 5th. S. 23° 15' E., to cut off 25 acres by a line running from the place of beginning and falling on the 5th side ; required the bearing and distance of the division. line. Ans. The division line bears from the place of beginning N. 45° 3' E. dist. 10.65 ch.

The following neat and concise method of solving this problem was furnished me by Robert Patterson, Professor of Mathematics in the University of Pennsylvania. As it requires the bearings of the sides to be changed so that one of them may become a meridian, it will be proper first to shew how this may be performed.

When the bearing of the side to be made a meridian is

north-easterly or south-westerly. Add the bearing of the side, made a meridian, to the north-westerly and south-easterly bearings ; if either of the sums exceed 90°, subtract it from 180°, changing N. W. to S. W. or S. E. to N. E.

But subtract it from the north-easterly and south-wesfrom which it is to be subtracted, take the difference, changing N. E. to N. W. or S. W. to S. E.

When the bearing of the side to be made a meridian is

north-westerly or south-easterly.

Add the bearing of the side, made a meridian, to the north-easterly and south-westerly bearings ; if either of the sums exceed 90°, subtract it from 180°, changing N. E. to S. E. or S. W. to N. W.

But subtract it from the north-westerly and southeasterly bearings; if it be greater than either of the bear. ings from which it is to be subtracted, take the difference, changing N. W. to N. E. or S. E. to S. W.

RULE 2.

Conceive the survey turned round, so that the side DE on which the division line AH is to fall may become a meri. dian, and change the bearings accordingly.

With the given distances and changed bearings find the corresponding latitudes and departures ; add together the numbers in each departure column, and take the difference of their sums, which will be the departure of the division line AH, and of the same name with the less sum ; place this departure in its proper place against the side AH, and beginning with it, find the meridian dis. tances according to the directions, given in rule 3, sect. 2, Measuring Land ; multiply the upper meridian distances by the corresponding latitudes as far as they are given, placing the products in the column of north or south area as directed in that rule; add together the pro

ducts in each of those columns, subtract the less sum from the greater, and take the difference between the remainder and double the area to be cut off; divide this difference by the upper meridian distance corresponding to the side DH, on which the division line is to fall, the quotient will be the latitude of the side DH, which place * against it in the column of north or south latitude according as its bearing is north or south ; add together the numbers in each latitude column, and take the difference of their sums, which will be the latitude of the division line AH, and of the same name with the less sum; then with the latitude and departure of AH, find, by trigonometry, its bearing and distance.

a

Change the bearing thus found by applying to it the angle expressing the bearing of the line, made a meridian, in a manner contrary to that in which it was applied in changing the original bearings, and it will give the proper bearing of the division line*.

EXAMPLES.

1. Let the bearing of AB be N. 62°1 W. 14.75 ch. BC, N. 19° E, 27 ch. CD, S. 77° E. 22.75 ch. and DE, S. 27° E. ; it is required to cut off 70 acres by a line AH, running from the angle A and falling on the side DE.

* The truth of this rule is too evident to need a demonstration.