(A—B) = 15° 55' 19'' log tan 9.455261 Having now }(A+B) and {(A—B), we find

A = }(A+B) + }(A−B) = 68° 23' 11.5'' and B = }(A+B) — (A—B) = 36° 32′ 33.5'' a sin C

whence

3. From (55), c =

whence c = 92.5.

sin A

to find the angles.

29. CASE 5. Given the three sides.

In the triangle ABC, given a = 120, b = 102 and c = 75,

1. Given A 30°, B=40° 50', and a=83, to find C, b and c.

2. Given a 50, b=40, and B=32°, to find A, C and c. 3. Given A=25° 30', B=50° 20′, and c=100, to find C, a and b.

4. Given A=32° 22' 12'', b=125, and c=212, to find B, C and a.

5. Given a 134, b=143, and c=314, to find A, B and C. 6. Given A=90°, a=234, and B = 40° 15', to find C, b

and c.

7. Given B=90°, b=84, and a=62.5, to find A, C and c. 8. Given C=90°, A=34° 45' 10'', and b=50, to find B, a and c.

9. Given A=90°, b=75, and c=45, to find B, C and a. 10. Given A=90°, a+b=207, and c=69, to find B, C, α and b.

11. Given C=29° 46', a-b=30, and sin A: sin B :: 7 : 4, to find A, B and c.

12. Given a+b+c=54, a−b=6, and b-c=6, to find A, B and C.

SECTION II.

DEFINITIONS AND INSTRUMENTS.

31. Surveying is the art of making the necessary measurements for determining the relative location of points on the surface of the earth.

It is of two kinds or branches-Plane and Geodesic. 32 Plane Surveying is that branch in which the curvature of the earth is disregarded.

It is limited to surveys of moderate extent, as of farms, town sites, etc.

33 Geodesic Surveying is that branch in which, for reason of the extent of surface comprised by the work, account is required to be taken of the curvature of the earth.

The terms Land Surveying, Railroad Surveying, Topographical Surveying, and some others, are used with reference to certain special purposes or practical objects to which the work is applied.

34. The operations comprised in Land Surveying may be classed under Field Work, Platting, and Computation of Area.

35. Field Work includes all operations required to be performed on the ground, as finding points which have been previously established, measuring to establish the location of other points, keeping memoranda, etc.

The memoranda constitute what are called the Field Notes.

The measurements usually required in Surveying are of two kinds-the measurement of distances or lines and the measurement of direction or angles. Accordingly, two classes of instruments are employed.

36. The principal instrument of linear measure, in Surveying, is the Chain.

The Steel Tape is, however, rapidly usurping the place of the chain. It is lighter and much more convenient to use. It is also a much more accurate measure than the chain.

The chain is 66 feet in length and consists of 100 links. Each tenth link is marked by a piece of brass, notched in a manner to indicate the number of tens from the end of the chain. Swivels are also inserted to prevent twisting. Steel tapes are made of various lengths. One of 66 ft., divided in links, is adapted to chain measurement.

37. Ten iron or steel pins, called Marking Pins, accompany the chain. They are usually about a foot in length and an eighth of an inch in thickness. They are pointed at one end for sticking in the ground, and bent into a ring at the other so as to be strung upon a ring for convenience of carrying.

Instead of ten pins, eleven are sometimes employed, and in place of a ring for carrying, a quiver is sometimes used.

38 Stakes with flags attached, and slender rods variously painted so as to be readily distinguished at a distance, are employed in ranging out long lines and marking points toward which measurements are required to be made.

39. Chain Measure Table:

7.92 inches make 1 link (1.)

Links are conveniently expressed as hundredths of a chain. Thus 51., 171., 8 ch. 121. may be written 0.05 ch., 0.17 ch., 8.12 ch. Any number of chains is reduced to feet by multiplying by 66, and, conversely, by dividing by 66. Thus, 5.13 ch.5.13 × 66 ft. = 338.58 ft., and 215 ft. 7 in. = 215.583 ft. = 215.583 ÷ 66 = 3.27 ch., nearly.

40. Chaining -Preliminary to any operation in Surveying, it is important to consider whether the instru

ments required in the work are in proper condition and adjustment for use. So simple an instrument as the chain forms no exception to this rule. It is liable to become lengthened by the wearing or opening of the links. or shortened by the links becoming bent.

It should, therefore, be frequently tested by comparing it with a carefully measured length of 66 feet, set off in a convenient place for use.

An error of length not readily remedied should be noted and the proper correction made of apparent measurements.

In measuring lines, marks should be set up at the extremities, and also at one or more intermediate points when the extremities are not visible from each other.

The operation of chaining will be better learned by practice than by precept. We, accordingly, omit any formal description of the process in this connection.

The following suggestions may, however, be serviceable:

In starting to measure a line, the forward chainman should see that he has the right number of marking pins, and that the chain or tape is free from kinks, and straight.

He should keep close to the line when chaining-never wandering off to either side. He must guide himself within narrow bounds by ranging forward or backward.

He should draw the chain taut and always at the same tension. It is sometimes more important that the measuring be uniform than that it be absolutely exact. The chain should be pulled steadily, never with a jerk, especially in going up hill.

The marking pins must be set plumb. The work should proceed briskly-no dawdling or gaping about is allowable. The forward chainman should not stop and look around until the rear chainman halts him.

In the counting of odd links, care must be taken not to count from the wrong end of the chain, giving 60 instead of 40, etc.; nor to count the wrong way from the tag, giving 38 instead of 42, etc. Too much care cannot be taken to avoid making mistakes.

The rear chainman should observe the condition of the chain as the head chainman pulls out.

It is for him also to see that the forward chainman keeps in line. He should call "Halt in time to avoid the chain being brought up with a jerk. He should not pull up the marking pin before the leader has set his in the right place and called out "Stuck." He should then step forward promptly, not requiring the leader to drag him in addition to the chain.