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The manner of calculating the area in this example is entirely similar to that explained in Art. 87. The meridian, from which the meridian distances are calculated, passes through station 7; the double meridian distances are all east. The half difference of the sums 5965629.35 and 4521985.79, is 721821.78: the number of square feet contained by the lines, Al, 1 2, 23, 34, 45, 56, 67, 7B, BC, CD, and DA.
We are next to compute the area which lies without the lines traced with the compass.
Beginning at station A, and regarding the lines which join the extremities of the offsets as right, which we may do without any sensible error, the area of the trapezoids
To compute the area of the quadrilateral legh.
Join 1 and g. Then, in the right-angled triangle lgh, there are known lh=140, and hg=40: the remaining parts can therefore be found ; they are, the angle glh=15° 56'43", the hypothenuse lg=145.62, and its area =2800.
But the bearing from A to 1 is N. 27° E., consequently, from 1 to A, S. 27° W. (168); and since le is perpendicular to Al, its bearing is known, viz. S. 63o E. The bearing of the line 1 2 is N. 111 E.; hence the angle hle is known, it being equal to 180° - (63° +11° 30)=105° 30°; and therefore the angle gle is known, being equal to 105° 30— 15° 56' 43"= 89° 33' 17". Therefore, the area of the triangle gle can be found, two sides and the contained angle being known; it is equal to 8007.9 square feet.
The areas of the quadrilaterals 2ikl, at station 2, and 3nmp at station 3, are computed in a similar manner by dividing them into the triangles 2ik, 2kl, and 3nm, 3mp; and the same for the quadrilaterals at stations 4, 5, 6, and 7.
The areas of the trapezoids and quadrilaterals will be arranged with reference to the lines to which they correspond, and for the sake of convenient reference, will be numbered from the beginning of those lines respectively.
Line Al. 1st Trapezoid=25000 2d Do. =27000 3d
Do. =10500 4th Do. =10400
Line 1 2. Quadrilateral = 10807.9 1st Trapezoid= 6500
Line 2 3. Quadrilateral = 5831.86 1st Trapezoid=10800 2d Do. =16100 3d Do. =21450
Line 3 4. Quadrilateral = 18766.85 1st Trapezoid=18700 2 Do. =16200 3d Do. =14400
Line 4 5. Quadrilateral =21184.3 1st Trapezoid=22500
Line 5 6.
= 7700 5th Do.
Line 7B. Quadrilateral = 7793.28 1st Trapezoid=17100 2d Do. =33712.50
193. We are yet to find the area of the ground surveyed with the plain table. This is done by dividing it into triangles, and measuring their bases and perpendiculars by means of the scale of equal parts. Thus, in the triangle BFM (the line BGM being nearly right), by applying the base BM to the scale of equal parts, it is found to be equal to 720 feet, and by applying the perpendicular Ff, it is found equal to 220 feet; and similarly, MQ=860, QW=250, BE, before found, =1623.65, EO=385, RN=470, OS=418, RS=310, TR=320, SS'= 130, and AA"=210. The area of the triangle BMF= 79200 square feet.
The area within the compass lines=72121is square feet
square feet in an acre, gives for the area. (110) 4 1., 2R, 32 P. Let this area,
41, ? R., 33 P., be added to the area ADCBEA =H 1., O R., 22 P., and we have
the total area
= 48 A., 3 R., 14 P.
194. Surveying being merely the application of the principles of geometry and trigonometry to the determination of the area of land, and the lengths and directions of lines, and such application varying with the kind of survey to be made, the form of the ground, and many other accidental circumstances, it is unnecessary, even were it possible, to follow the surveyor to every case that may arise, and give him a rule precisely applicable to it. In operations which run out into such a variety of detail, general principles only can be dwelt upon; hypothetical cases would be uninteresting, and might be useless.
The experienced surveyor will regard his theodolite with peculiar interest. It is the great instrument. It alone can be relied on for nice and accurate operations.
A large section of country may be accurately surveyed, by determining a system of consecutive triangles. In such a survey, stations should be chosen on the tops of hills or mountains, and at other prominent and important points; and this general rule ought never to be neglected—measure the three angles of every triangle, whenever it is possible ; this will prove the work as it advances.
The tracing of the shores of rivers, creeks, roads, fences, &c., may safely be trusted to the compass, after having determined several of their prominent points by triangulating the ground with the theodolite. The compass, however, is a rude
instrument, and cannot be relied on, unless used in conjunction with the theodolite. It is, nevertheless, used to determine the area of ground by the methods explained in Chapter V.; yet, if the land were valuable, greater accuracy would certainly be necessary
The plain table is used to great advantage when only a plot of the ground is wanted. It ought not to be used for the determination of long lines, nor can it be relied on in determining extended areas.