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To protract the survey.
Draw NS, Fig. 75, to represent a meridian line; then N standing for the north and S for the south, the east will be to the right hand and the west to the left. In -NS take any convenient point as A for the place of be ginning, and apply the straight edge of the protractor to the line, with the centre to the point A, and the arch turned toward the east, because the first bearing is easterly; 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 easterly 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 line, 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
With the chord of 60° describe the circle NESW, Fig. 76, and draw the diameter NS. Take the several bearings from the line of chords, and lay them off on the circumference from N or 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 1 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.
To find the area.
By drawing lines as in Fig. 75, the survey is divided into two trapeziums AGFE, AEDB, and a triangle BDC. Measure the several bases and perpendiculars, on the same scale that was used in the protraction, and find the double areas of the triangle and trapeziums by probs. 2 and 6; the sum of these will be the double area
of the survey.
EG 16.68 x
EB 19.17 X
=267.4215 = 2 AEDB
BD 19.23 x Ce 5.16 = 99.2268 = 2 BDC
The following field-notes are given to protract the sur
It is required to protract the survey and find the area from the following field-notes.
Containing three different rules for finding the areas of right-lined figures generally, when the bearings and distances of the boundaries are given.
1. Meridians are north and south lines, which are supposed to pass through every station of the survey.
2. The difference of latitude, or the northing or southing of any stationary line, is the distance that one end of the line is north or south from the other end; or it is the distance which is intercepted on the meridian, between the beginning of the stationary line and a line drawn from the other end, perpendicular to that meridian. Thus, if NS, Fig. 77, be a meridian passing through the point A, of the line AB, then is Ab the difference of latitude, or southing of that line.
3. The departure of any stationary line, is the nearest distance from one end of the line to a meridian passing through the other end. Thus Bb, Fig. 77, is the departure or easting of the line AB. But if ns be a meridian, and the measure of the stationary line be taken from B to A, then is BC the difference of latitude, or northing, and AC the departure or westing of the line AB.
4. The meridian distance of any station, is the distance thereof from a meridian passing through the first, or some other particular station of the survey.
5. The Traverse Table is a table containing the dif ference of latitude and departure corresponding to dif
To find the difference of latitude and departure corresponding to any given course and distance, by means of the annexed Traverse Table.
When the distance is any number of whole chains or perches not exceeding 100.
Find the given bearing at the top or bottom of the table according as it is less or more than 45°. Then against the given distance, found in the column of distances at the side of the table, and under the bearing, if at the top, or over it if at the bottom, is the corresponding difference of latitude and departure. The difference of latitude and departure must be taken as marked at the top of the table when the bearing is at the top, but as marked at the bottom, when the bearing is at the bottom. Thus if the bearing and distance be S. 35° 15′E. dist. 79Ch., the diff. of lat. will be 64.51Ch. S. and the dep. 45.59Ch. E.: but if the bearing and distance be S. 54° 45'E. dist. 79Ch. the diff. of lat. will be 45.59Ch. S. and the dep. 64.51.Ch. E.
When the distance is expressed by any whole number of chains or perches exceeding 100.
Divide the given distance into parts that shall not exceed 100 each, and find as before the difference of latitude and departure corresponding to the given bearing and to each of those parts; the sums of the latitudes* and departures thus found will be the latitude and de. parture required.
For the sake of conciseness in the expression the word latitude only is