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the plates DLC and AIB, the use of which will be explained hereafter.
Near the two other edges of the table, two small grooves are made, into which the plates of brass DB and CA are fitted, and these plates are drawn to their places by means of milled screws which pass through the table from the under side, and screw firmly into the plates. The heads of two of the screws, Q and S, are seen in the figure, as also one of the plates and its two screws in Fig. 3. The object of these plates is to confine a sheet of paper on the table. By loosening the screws, and pressing them upwards, the plates are raised above the surface of the table; the edges of the paper can then be placed under them : then, by turning the screws back again, the plates are drawn down and the paper held tightly. Fig. 1 represents the table with the paper partly put upon it: one edge of the paper has been placed under the plate DB, and the screws S and Q, tightened. The paper, before being put on, should be moistened, in order to expand it; and then, after it has been dried, it will fit closely to the table.
A ruler, AB (Fig. 2), with open vertical sights, is used with the plain-table. This ruler has a fiducial edge, which is in the same vertical plane with the hairs of the sights. A ruler with a telescope, and a vertical limb, similar to the vertical limb of the theodolite, is sometimes used with the plain-table. A compass, also, is often attached to the table, to show the bearings of the lines.
The plain-table is used for two distinct objects. 1st. For the measurement of horizontal angles.
2dly. For the determination of the shorter lines of a survey, both in extent and position.
To measure a horizontal angle. 161. Place, by means of a plumb, the centre of the table directly over the angular point: then level the table ; after which, place the fiducial edge of the ruler against the small pin at the centre : direct the sights to one of the objects, and note the degrees on the brass plate; then turn the ruler and sights to the other object, and note the degrees as before. If the ruler has not passed over the point, the difference of taken from 180°, and the remainder added to the smaller, gives the required angle.
Of the determination of lines in extent and position. 162. Having placed a paper on the table, examine the objects and lines which are to be determined, and measure a base line in such a direction, if possible, that all the objects can be seen from its extremities. Then place the plain-table with its centre, nearly, though not accurately, over one extremity of the base; make it truly horizontal, and turn it until the larger part of the paper lies on the same side of the base with the objects.
Then, tighten the clamp-screw, and mark with a pin the point of the paper directly over the station, which point is determined most accurately by suspending a plumb from the lower side of the table. Press the pin firmly on this point, bring the fiducial edge of the ruler against it, and sight to the other extremity of the base line, and mark with the pin or pencil, the direction of the line on the paper. Sight in like manner to every other object, and draw on the paper the corresponding lines, numbering them from the base line, 1, 2, 3,
Then, with a pair of dividers, take from the scale a certain number of equal parts to represent the base, and lay off the distance on the base line from the place of the pin. Take up the table, carry it to the other extremity of the base, and place the point of the paper corresponding to that extremity, directly over it. Place the fiducial edge of the ruler on the base line, and turn the table, by means of the tangent-screw, until the sights are directed to the first station. If, however, in bringing the table to this position, the corresponding point of the paper has been moved from over the extremity of the base line, move the legs of the tripod until it is brought back to its place. Let the table be then levelled, after which, place the ruler again on the base line, and bring the table to its proper position by the tangent-screw, and continue the adjustment until the extremity of the base line on the paper is directly over the station, and in the same vertical plane with the base line on the ground. Then direct the sights to all the objects sighted to from the other station, and mark the
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,
A which we will suppose equal to 300 yards. Place the plaintable at A, and sight to the 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 i 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.
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
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.
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
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 truc levels of those points.
But AB=BE (BE+2EC), and ADP=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
inco: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 Ar=DFX 2 FC, AE
AF ; and DF=
2FC and since the diameter of the earth is constant, BE and DF are proportional to AE and AF.
* The spheroidal form of the earth is not considered, as it affects the results