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1. How many acres are in a circle a mile in diameter ?
5026.5600 Sq. Ch. 50 2A. 2R. 25P. nearly.
2. Required the area of an ellipsis, the longer diameter of which measures 5.36ch. and the shorter 3.28ch.
3. Required the area of a circular park, the diameter of which is 100 perches. Ans. 49A. OR. 14P,
4. Required the area of an elliptical fish pond, the longer diameter of which is 10 perches, and the shorter 5 perches. Ans. 39.97 Sq. Per.
To protract a Survey, and to find its area by dividing it into triangles and trapeziums.
The method of doing this will be best understood by an example. Thus,
Suppose the following field-notes to be given, it is required to protract the survey and find its area.
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 beginning, 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 (or not as may be) 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 arcas of the triangle and trapeziums by probs. 2 and 6; the sum of these will be the double area of the survey.