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The area within the
compass lines=721821.78
The area without the compass lines=534211.49
The area found with the plain table=791421.25

Sum, 2047454.52.

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This sum being divided by 43560, the number of square feet in an acre, gives for the area, (110) 4 A., 2 R., 32 P.

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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

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.

CHAPTER VII.

OF LEVELLING.

195. Ir all the points of the earth's surface were equidistant from the centre, it would be perfectly even, and present to the eye an unbroken level.

Intersected, however, as it is, by valleys and ridges of mountains, it becomes an important problem to ascertain the difference between the distances of given points from the centre of the earth; such difference is called the difference of level; and a line, all the points of which are equally distant from the centre, the line of true level.*

196. One point is said to be above another, when it is farther from the centre of the earth; and below it, when it is nearer.

197. Let C (Pl. 9, Fig. 1,) represent the centre of the earth, A a point of its surface, and AEF the line of true level. If, at the point A, a tangent line GABD be drawn to the surface, such line is called the line of apparent level.

198. Now, if an instrument were placed at A, which could be brought into a horizontal position to indicate a horizontal line, this line would be tangent to the earth at A, and would be the line GABD of apparent level.

199. When, therefore, we have ascertained the direction of a tangent, or horizontal line, we have found the line of apparent level only; the line of true level is yet to be determined.

If at the points E and F vertical staves be placed, the line of apparent level passing through A will cut them at B and D, while the line of true level cuts them at E and F.

* The spheroidal form of the earth is not considered, as it affects the results too inconsiderably to be regarded in the common operations of levelling.

Therefore, BE and DF are, respectively, "the differences between the apparent levels of the points E and F, as determined by the horizontal line passing through A, and the true levels of those points.

=

But AB BE (BE+2EC), and AD2=DF (DF+2FC) (228)*. In the common operations of levelling, the arcs AE, AF, are small; and since the difference between small arcs and their tangents is very inconsiderable, the arcs AE, AF may be substituted for the tangents AB, AD. And since the external parts of the secants BE and DF are very small in respect of the diameter of the earth, they may be neglected without sensible error: the expressions above will then become, AE BEX2EC, and AF2 =DF x2FC,

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and since the diameter of the earth is constant, BE and DF are proportional to AE2 and AF2.

But BE and DF are respectively the differences between the true levels of the points E and F, and their apparent levels, as ascertained from the point A: hence, the difference between the apparent and true level of any point, is equal to the square of the distance of that point from the place where the apparent level was made, divided by the diameter of the earth; or, the diameter being constant, the rise of the apparent above the true level, is proportional to the square of the distance.

200. The mean diameter of the earth being about 7921 miles, if AE be taken equal to 1 mile, then, the excess BE= becomes equal to -=8.001 inches.

AE2 2AC'

1

7921

If the excess FD, for any other distance AF, were required, AE2: AF2: BE: FD; and by similar proportions, the fol

Table showing the differences in inches between true and apparent level, for distances between 1 and 100 chains.

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6.846

99

12.246

24 .720 49 3.004 74
25 .781 50 3.125 75 7.032 100 12.502

We cannot proceed farther in the discussion of the principles of levelling, until we have described the instruments which are to be used, and explained the particular objects that they are to

answer.

OF THE LEVEL.

201. The level is an instrument used to determine horizontal lines, and the difference of level of the different points on the

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