sections of the corresponding lines 1,1, 2,2, 3,3, 4,4, &c., determine, on the paper, the positions of the several objects; and a reference of these lines to the scale of equal parts, determines the true distances.

163. Let it be required, for example, to determine, by means of the plain-table, the relative position of several houses.

Measure the base line AB, which we will suppose equal to 300 yards. Place the plaintable at A, and sight to the

A

corners of the houses, and mark the lines 1, 2, 3, 4, &c. Then remove the table to B, and sight to the same corners as before, and draw the lines as in the figure. The points at which they intersect the corresponding lines before drawn, determine the corners of the houses. The front lines of the

houses may then be drawn on the paper. Draw lines at right angles to the front lines, and on them lay off the depths of the houses, with the same scale as that used for the base line

To find the length of any line drawn on the paper, as the line 1, drawn through A, for example, place the dividers at A and extend them to the other extremity of the line, and then apply the line to the scale. The length of the line 1 is equal to 198 yards.

164. In this example, we determine from the base line CD, the positions of the points B, F, E, and H.

E

F

હ

2

H

B

Of changing the Paper.

165. When one paper is filled, and there is yet more work to be done, let the paper be removed, and a second paper put on the table; after which, the table may be used as before.

Now, in order that the two papers may be put together and form one entire plan, it is necessary that two points determined on the first paper, be also determined on the second; and then, by placing the lines joining these points upon each

relative position as the corresponding lines on the ground; and the same for as many papers as it may be necessary to use. If different scales are used, the corresponding points will not join, and then the work must be reduced to the same scale, before the papers can be put together.

In the first example, the position of the point F was determined, in order to unite the first paper with the second.

In the second example, we sighted from C and D, the extremities of the base line, to the points B and F; we thus determined the line BF on the second paper. Placing the line BF of the one paper on BF of the other, we have the following plan.

In this plan, all the points and lines are accurately laid down. Any number of papers may be joined in the same

manner.

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 V.

Of Levelling.

166. If 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, is called the line of true level.*

167. 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.

168. Let C (Pl. 4, 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 ABD be drawn to the surface, such line is called the line of apparent level.

169. Now, if an instrument, were placed at A, and brought into a horizontal position so as to indicate a horizontal line, this line would be tangent to the earth at A, and would be the line ABD of apparent level.

170. 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. 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 AB2=BE (BE÷2EC), and AD2=DF (DF÷2FC) (Geom. Bk. IV, Prop. XXX). In the common operations of levelling, the arcs AE, AF, are small; and since the difference between small arcs and their tangents is very incon siderable, 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 comparison with the diameter of the earth, they may be neglected without sensible error. the expressions above will then become,

AE

BEX2EC, and AF2DFx2FC,
AE2
2EC

or, BE. ; and DF=

AF2

2FC

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

* The spheroidal form of the earth is not considered, as it affects the results

But BE and DF are respectively the differences between the true levels of the points E and F, and their apparent levels, as determined 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.

171. The mean diameter of the earth being about 7919 miles, if AE be taken equal to 1 mile, then the excess

AE2 BE: = 2AC

becomes equal to

1

7919

8.001 inches.

If the excess FD, for any other distance AF, were required, AE AF :: BE: FD;

:

and by similar proportions the following table is calculated.

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

Chains. Inches. Chains. Inches. Chains. Inches. Chains.

Inches.

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 which they are to answer.

OF THE LEVEL.

172. The level is an instrument used to determine horizontal lines, and the difference of level of any points on the surface of the earth.

The part of the instrument shown in Pl. 4, Fig. 2, rests on a tripod to which it is permanently attached at Z. HH is a horizontal brass plate, through which four levelling screws with milled heads are passed, and worked against a second horizontal plate GG. Two of these screws, K and I, are seen in the figure. S is a clamp-screw, which, being loosened, allows the upper part of the instrument to turn freely around its axis. Q is a tangent-screw, by means of which the upper part of the instrument is moved gently, after the clamp-screw S has been made fast. EE is a horizontal bar, perpendicular to which are the wyes, designated Y's, that support the telescope LB. This telescope is confined in the Y's by the loops r, r, which are fastened by the pins p and p. The objectglass B, is adjusted to its focus by the screw X; the eyeglass L slides out and in freely. The screws f, f, work the slide which carries the horizontal hair; and two horizonta! screws, only one of which, a, is seen, work the slide that carries the vertical hair. CD is an attached spirit level. The screw N elevates and depresses the Y, nearest the eye-glass. In some instruments this Y is elevated and depressed, by means of two screws at M and R.

Before using the level, it must be adjusted. The adjustment consists in bringing the different parts to their proper places.

The line of collimation is the axis of the telescope. With this axis, the line drawn through the centre of the eye-glass, and the intersection of the spider's lines, within the barrel of the telescope, ought to coincide.

FIRST ADJUSTMENT.* To fix the intersection of the spider's lines in the axis of the telescope.

Having screwed the tripod to the instrument, extend the

* This, and some of the following adjustments, are so similar to those of the theodolite, that they would not be repeated, but that some may use the level without wishing to study a more complicated instrument.