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the number of outs, the distance measured is readily determined.

All slant or inclined surfaces, as the sides of a hill, should be measured horizontally, and not on the plane or surface of the hill. To effect this, the hind end of the chain, in ascending a hill, should be raised from the ground till it is on a level with the fore end, and, by means of a plummet and line, or when the hill is not very steep. by estimation, should be held perpendicularly above the termination of the preceding chain. In descending a hill, the fore end of the chain should be raised in the same manner, and the plummet being suspended from it will show the commencement of the succecding chain.

PROBLEM VII. To protract a Survey, having the bearings and distances

of the sides given. The method of doing this will be best understood by an example. Thus,

Suppose the following field-notes to be given, it is re quired to protract the survey.

Ch. 1. N. 50° E. 9.60 2. S. 32° E. 16.38 3. S. 41° W. 6.30 4. West 8.43 5. N. 79° W. 10.93 6. N. 5° E. 11.25 7. S. 83° E. 6.48

Method Ist. Draw NS, Fig. 75, to represent a meridian line; then N standing for the north and S for the south, the cast

NS take any convenient point as A for the place of beginning, and apply the straight edge of the protractor to the line, with the centre to the point A, and the arch turned towards the east, because the first bearing is casterly; then holding the protractor in this position, prick off 50° the first bearing, from the north end, because the bearing is from the north ; through this point and the point A, draw the line AB on which lay 9.60 chains, the first distance from A to B. Now apply the centre of the protractor to the point B, with the arch turned toward the east, because the second bearing is easterly, and move it till the line AB produced, cuts the first bearing 50°; the straight edge of the protractor will then be parallel to the meridian NS; hold it in this position, and from the south end prick off the second bearing 32°; draw BC and on it lay the second distance 16.38 chains. Proceed in the same manner at each station, observing always, previous to pricking off the succeeding bearing, to have the arch of the protractor turned casterly or westerly according to that bearing, and to have its straight edge parallel to the meridian; this last may always be done by applying the centre to the station point, and making the preceding distance lipe produced if necessary, cut the degrees of the preceding bearing. It may also be done by drawing a straight line through each station, parallel to the first meridian.

When the survey is correct, and the protraction accurately performed, the end of the last distance will fall on the place of beginning.

Method 2nd.

With the chord of 60° describe the circle NESW, Fig. 76, and draw the diameter NS. Take the several bear

ings from the line of chords, and lay them off on the circumference from Nor S according as the bearing is northerly or southerly, and towards E or W according as it is easterly or westerly, and number them 1, 2, 3, 4, &c. as in the figure. From A the centre of the circle, to I draw A 1, on which lay the first distance from A to B; parallel to A 2 draw BC, on which lay the second distance from B to C: parallel to A 3 draw CD, on which lay the third distance from C to D; proceed in the same manner with the other bearings and distances.

EXAMPLE 2.

The following field notes are given, to protract the survey.

Ch.
1. N. 15° 00' E. 20
2. N. 37° 30' E. 10
3. East 7.50
4. S. 11° 00' E. 12.50
5. South 13.50
6.
West

10.
7. S. 36° 30' W. 10.
8. N. 38° 15' W. 8.50

PROBLEM VIII.

The bearing of two lines from the same station being given,

to find the angle contained between them.

RULE.

When they run from the same point of the compass, towards the same point, subtract the less from the greater.

When they run from the same point, towards different

When they run from different points, towards the same point, add them together, and take the supplement of the sum.

When they run from different points, towards different points, subtract the less from the greater, and take the supplement of the remainder.

Note.- When the bearing of one of the lines is given towards the station, instead of from it, take the reverse bearing of such line; the angle may then be found by the above rule.

EXAMPLES.

1. Given the bearing of the line AB, Fig. 67, V. 34° E., and AD, N. 58° E. ; required the angle A.

AD, N. 58° E.
AB, N. 34° E.

Angle A=24°

2. Given the bearing of BA, Fig. 57, S. 34° W., and BC, S. 35° E. ; required the angle B. Ans. B=69°

3. Given the bearing of BC, Fig. 67, S. 35° E., and CD, S. 87° W.; required the angle C. Ans. 58o.

4. Given the bearing of DC, Fig. 67, N. 87° E., and DA, S. 58° W.; required the angle D. Ans. 151°

PROBLEM IX.

To change the bearings of the sides of a survey in a cor

responding manner, so thut any particular one of them may

become a Meridian.

RULE.

Subtract the bearing of the side that is to be made a meridian, from those bearings that are between the same points that it is, and also from those that are between points directly opposite to them. If it is greater than either of the bearings from which it is to be subtracted, take the difference, and change E. to W., or W. to E.

Add the bearing of the side which is to be made a meridian, to those bearings which are neither between the same points that it is, nor between the points that are directly opposite to them. If either of the sums exceeds 90°, take the supplement and change N. to S., or S. to N.*

Note.—When the bearings of some, or all, of the sides of a survey have been thus changed, and by calculation the changed bearing of another side or line has been

* The changing of the bearings so as to make a given side become a meridian, may be illustrated by means of a protracted survey. If a protracted survey or plot is held horizontally, with the meridian in a north and south direction, the north end being towards the north, the bearings of the sides of the plot will then correspond with the bearings of the sides of the survey. If then, keeping the paper horizontal, it be turned round till any particular side of the plot has a north and south direction, or becomes a meridian, the bearings of all the other sides of the plot will have been changed by a like quantity. But it is evident, that neither the relation of the different parts of the plot to one another, the area nor the lengths of the sides will have been altered by this change. It may be here observed, that some calculations in surveying are considerably shortened by changing the bearings so as to make a certain side become a meridian. The method was communicated to me by Robert Patterson, late Professor of Mathematics and Natural Philosophy

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