Area, 1909,11 Chains, or 190 Acres 3 Roods 25 Rods.

Having protracted this Field divide the Map into its appropriate number of Triangles, and calculate their Area as directed in the last Problem. The number of Triangles into which a Field is divided will always be two less than the number of its Sides.

NOTE. The Links operate as Decimal parts of a Chain.

EXAMPLE III.

In the preceding Examples, the corners of the Field are supposed to be visible from one to the other, or the Courses of the Lines known; but it is frequently the case, that the Surveyor cannot avail himself of either of these advantages; in such cases, the following method will be found accurate, easy and expeditious.

Fig, 48,

D

E

A

B f

The Survey of the Field ABCDE, (See Fig. 48.) is begun at Station A. The corner at B cannot be seen from A, nor is the Course of the Line known. A Course S. 3° E. is taken from A towards B, represented by the dotted line Af; then "Measure the nearest Distance Bf, from the Corner, to the Random Line Af,"* which may be called the Stationary Distance, and in this Example is 38 Links. The length of the Random Line, to this place, will be the true Distance of the Line AB, which is 44 Rods. To calculate the true Course of the Line AB, use the following Proportion. As the length of the Line run, 44 Rods, is to 57,3† so is the Stationary Distance 38 Links, to the Variation, necessary to run the true Course. This must be added to, or subtracted from, the Course of the Random Line, as the case may require, and the true Course is obtained; thus,

Rods Deg. Links

As 44: 57,3 :: 38: 1° 58′ 7′′ or 2° nearly.

Subtract 2° from the Course of the Random Line, and it will leave 1°; therefore, S. 1°E. 44 Rods must be entered as the first Line of the Field Book. The course of the Line BC is known; N. 76° W. its distance is taken to 2, on the bank of a pond, from whence Measure the the corner at C, on the opposite bank, can be seen, Line 2, 3, and take its bearing N. 4° E. 8 Rods; take the bearing of 3 C, S. 80° W. there is then, in the Triangle 2 C 3, the Angles, known by the Bearings, and one Side given, to find 2 C. This is performed by Case 1. Oblique Trigonometry. An account of the Triangle must be inserted in the Field Book, under the Second Line; and before the plan is drawn, the Distance 2 C must be added to 24 Rods, the Distance of B 2, and the Distance from B to C will be obtained. The Corner at D is a Tree, standing on a high ledge of rocks. The length of the Lines CD, and DE, cannot be measured with a Chain. Take the Course of the Line, from C to D, which note in the Field Book; then take the Course and Distance of a Line from C to E; N 2° W 35 Rods; then take the Course of the Line from E to D, S. 40° W. which insert in the Field Book. The Course and Distance, of the Line from E to A, is obtained in a similar way, as that from A to B, and is S. 88° E. 42 Rods.

* This nearest Distance should be taken nearly at Right Angles from the Random Line to the Bound. The Angle, however, should be as much less than a Right Angle, as half the quantity of the Angle at the first Station, made by the Random and true Line; which would make the Angles at B, and f, of the Triangle AfB, equal. (See Remark 1. Sec. 3. Part 1.)

† “57,3 is the Radius of a Circle (nearly) in such parts, as the Circumference contains 360,"

The true Course, as found by this method, will seldom differ two minutes of a Degree, from a statement to find the Course by Trigononretry.

D.

In the first place, complete the Triangle 2 C 3, by finding the contained Angles, at 2 and 3, (See Problem 1, of Distances, Sec. 4, Part 1.)

Add

2 C. 76° W.

39 N. 760 By Rule 1.

By Contracting the Triangle, the Distance 2C, across the Pond, is found 19 Rods, which makes the Line BC 43 Rods.The plan may then be drawn as directed in the preceding example, by laying off the Courses from Parallel Lines; or, more accurately, by finding the number of Degrees, in the contained Angles, made by the Bearing of the Lines. Under the Problem, last refered to, will be found the Rules for finding these Angles. Draw the Line AB, according to the directions before given, for laying of the first Course and Distance. Reverse the Course AB. and it will be N. 1° W. and the Course of BC is N. 76° W. of course, the Angle at B is 75°, found by the second Rule. Make an Angle at B of 75°, and draw BC in length 43 rods. Having plotted to C, the third Station, reverse the last Course, which must always be done to find the quantity of the Angle by these rules,* and find the Angle ECB, and lay off the Line CE; next find the quantity of the Angles DCE, and DEC, and lay them off from each

*The practitioner is supposed to stand at the Angular Point, and take the Bearings of the Lines; by which it is easy to perceive that the last Course plotted, as taken in the Survey, must be reversed.

end of the Line CE. The Intersection of these lines, at the Point D, represents the Corner on the Ledge. The Length of the Lines CD and DE, if required, may be measured in the Dividers on the Scale. Lastly, find the Quantity of the Angle AEC, or AED, and draw EA. Divide the plan into thrée Triangles, or one Triangle and one Trapezium; and calculate its Area as before directed.

This methed of Protracting a Field is preferable to that of doing it by Parallel Lines; it being difficult to draw them with perfect accuracy. An attention to the Courses will show in what Direction, the Angle is to be made. If there be an external Angle to the field, the quantity of the Angle, as found by these Rules, will be without the Field, and the Lines must be drawn accordingly.

EXAMPLE IV.

Fig. 49.

B

E

D

f

The survey of the Field ABCD was begun at A. The Lines AB, BC, and CD were surveyed according to the directions in the preceding examples. The Line DA passes through the Border of a Swamp thickly covered with bushes and other impediments, to avoid which, a Course and distance were taken from D to e, thence to f, and from thence to A.

calculate the Area of The Course and Dis

Having Protracted the Field according to the Survey, draw a Line from the third to the first Station, and that part only contained north of this Line. tance of this Line, N. 72° W. 20° 87 Links, may easily be ascertained on the plan.

To avoid impediments of this kind it is often practised to make an Offsett, as from D, at Right Angles from the Line DA, and keeping that Course till directly opposite the bound at A; but in following this method the Course of the Line DA ought to be known. It is likewise sometimes practised to consider such, as Closing Lines, and find their Course and Distance by the Protraction; but this method cannot be depended on, and prevents the detection of any error, which may have been committed in the survey.

To ascertain at what part of a survey an error was committed without the trouble of an entire resurvey, take a Course from every corner of the Field while performing the survey, to some elevated object therein, as at E, or, from as many corners as the object is visible, and insert the Courses on the left side of the Field Book, opposite the Stations from which they were respectively taken. In protracting, lay off each Course as they were taken; so far as these Lines intersect, or meet in one Point, all is right; but if one Line diverge from the point of Intersection, an error must have been committed on the Line preceding the Station, from which the diverging Line of Intersection was taken; so that by going to this part of the Field, the error can be readily corrected; care must however be taken, that no error be committed in Protracting, or taking the Courses of Intersection.