latitude and departure; balance the work, and find the double meridian distances.

2. Given the following field notes:

1st, N. 45° W., 20 ch.; 2d, N. 18° E., 12.25 ch.; 3d, E., 12.80 ch.; 4th, N. 32° E., 6.50 ch.; 5th, S. 421° E., 13.20 ch.; 6th, S., 14.75 ch.; 7th, S. 654° W., 16.30 ch. : Required the corrected latitude and departure, and the double meridian distances.

Note 1.-It should be remembered that in finding the area of a tract of land the inequalities of its surface are not considered, but the tract is treated as a horizontal plane.

Note 2.-The area of a portion of land can, in a great variety of cases, be calculated by the rules already given for Mensuration of Plane Surfaces.

298. Problem.

To find the area of a tract of land when the length and direction of the bounding lines are given.

It is evident from the diagram that the area of ABCD is equal to the sum of the trapezoids EBCX and XCDH, minus the sum of the triangles AEB and ADH; and that twice the sum of the trapezoids, minus twice the sum of the triangles, is equal to twice. ABCD.

E

Q

P

Ꭺ

T

X

Y

H

B

R

W

C

The following table will exhibit the general form of operation:

Sta. Cour. NLat. SLat. DMD. Triangles. Trapezoids.

It will be observed that we have taken the most westerly station for the principal station, and have multiplied the double meridian distance of each course by its latitude, and that the product is double the area of a triangle when the latitude is north, and double the area of a trapezoid when the latitude is south.

If we had taken the most easterly station for the principal station, the reverse would be true.

In the above we have supposed that the lines were run in such direction as to keep the lot at the right.

If the lines were run in the opposite direction, so as to keep the lot at the left, the reverse would be true.

In any case, the sum of the double areas of the trapezoids, minus the sum of the double areas of the triangles, is equal to double the area required.

299. Rule.

Multiply the double meridian distance of each course by its latitude, placing the product in one column when the latitude is north, and in another column when the latitude is south, and divide the difference of the sums of the two columns by 2, and the quotient will be the area required.

Take the example of a preceding article whose D. M. D.'s have been found.

Sta. Bearings. Dist. NLat. SLat. EDep. WDep. DMD. Triang. Trap.

Divide double the area by 2, the result by 10 to reduce the chains to acres, multiply the decimal by 4 to reduce to roods, and the next decimal by 40 to reduce to perches.

2.907500

40

36.300000

300. Plotting.

Plotting is the process of representing, to a given scale, the length, direction, and relative position of the bounding lines of a tract of land.

1st Method. By means of latitudes and departures.

Take the example of the last article.

Let NS represent the meridian passing through the principal station A.

Select a scale whose unit shall represent 1 ch., and take AE 13.12 ch., the lat. of first course.

=

E

N

Through B draw a meridian, and take BF=7.10, the lat. of second course.

Through F draw a line perpendicular to BF; take FC 4.05 ch., the dep. of second course, and draw BC. Through C draw a meridian, and take CG 13.05, the lat. of third course.

=

=

Through G draw a line perpendicular to CG, and take GD 8.10 ch., the dep. of third course, and draw CD. Through D draw a meridian, and take DI=7.03 ch., the lat. of fourth course.

=

Through I draw a line perpendicular to DI; take IA 12.69 ch., the dep. of fourth course, and draw DA. Remark 1.-If the departure of fourth course terminates at A, the work will be verified.

2. It will be observed that N. lat. is laid off upward, S. lat. downward, E. dep. to the right, and W. dep. to the left.

3. The auxiliary lines can be drawn with a pencil/ and afterward erased.

4. If every scale in possession of the surveyor should make the diagram too large or too small, all the latitudes and departures can be divided or multiplied by the same number, and the results taken instead of the given latitudes and departures.

2d Method. By means of bearings and distances.

Take the same example.

Let NS represent the meridian passing through the principal station A.

With a protractor lay off the angle NAB 52°, the bearing of A first course, and take AB=21.28

ch., the first course.

Through B draw a meridian,

N

and lay off S'BC= 293°, the bearing of second course, and take BC : 8.18 ch., the second course.

Through C draw a meridian, and lay off S"CD=313°, the bearing of third course, and take CD 15.36 ch., the third course.

=

Through D draw a meridian, and lay off N'DA = 61°, the bearing of fourth course, and take DA 14.48 ch., the fourth course, which will terminate at A if the work is correct.

Remark 1.-The latitude and departure letters indicate the general direction of the lines, and the degrees the exact direction.