ence of the columns of Latitude will be the length of that wanting side which had been made a Meridian.

Thus, let the lengths of BC and DE be wanting, as in the pre

of DE, since BC, being a Meridian, has no Departure. Hence the length and Latitude of DE are readily obtained. This Latitude being put in the table, and the columns of Latitude then added up, their difference will be the length of BC.

CASE 4. When the Bearings of two sides are wanting.

(447) When the deficient sides adjoin each other. Find the Latitudes and Departures of the other sides, and then, as in Case 1, find the length and bearing of the line joining the extremities of the deficient sides. Then in the triangle thus formed we have the three sides to find the angles and thence the Bearings.

C

Fig. 306.
D

(448) When the deficient sides are separated from each other. Change the places of the sides so as to bring the deficient ones next to each other. Thus, in the figure, supposing the Bearings of CD, and EF to be wanting, transfer EF to DG, and DE to GF. Then calculate, as in Case 1, the length and Bearing B of the line joining the extremities of the deficient sides, CG in the figure. This line and the deficient sides form a triangle in which the three sides are

A

E

F

given to determine the angles and thence the required Bearings.*

* The fullest investigation of this subject, developing many curious points, will be found in Mascheroni's "Problemes de Geometrie pour les Arpenteurs," and Lhuillier's "Polygonometrie."

PART VIII.

PLANE TABLE SURVEYING.

(449) THE Plane Table is in substance merely a drawing board fixed on a tripod, so that lines may be drawn on it by a ruler placed so as to point to any object in sight. All its parts are mere additions to render this operation more convenient and precise.

Such an arrangement may be applied to any kind of "Angular Surveying"; such as the Third Method, "Polar Surveying," in its two modifications of Radiation and Progression, (characterized in Art. (220)), and the Fourth Method, by Intersections. Each of these will be successively explained. The instrument is very convenient for filling in the details of a survey, when the principal points have been determined by the more precise method of "Triangular Surveying" and can then be platted on the paper in advance. It has the great advantage of dispensing with all notes and records of the measurements, since they are platted as they are made. It thus saves time and lessens mistakes, but is wanting in precision.

(450) The Table. It is usually a rectangular board of well seasoned pine, about 20 inches wide and 30 long. The paper to be drawn upon may be attached to it by drawing-pins, or by clamping plates fixed on its sides for that purpose, or by springs pressed upon it, or it may be held between rollers at opposite sides of the table. Tinted paper is less dazzling in the sun. Cugnot's joint, described on page 134, is the best for connecting it with its tripod, though a pair of parallel plates, like those of the Theodolite, are often used. A detached level is placed on the board to test its horizontality; though a smooth ball, as a marble, will answer the same purpose approximately.

A pair of sights, like those of the compass, are sometimes placed under the board, serving, like a "Watch Telescope,” (Art. (339), to detect any movement of the instrument. To find what point on the lower side of the board is exactly under a point on the upper side, so that by suspending a plumb-line from the former the latter may be exactly over any desired point of ground, a large pair of "callipers," or dividers with curved legs, may be used, one of their points being placed on the upper point of the board, and their other point then determining the corresponding under point; or a frame forming three sides of a rectangle, like a slate frame, may be placed so that one end of one side of it touches the upper point, and the end of the corresponding side is under the table precisely below the given point, so that from this end a plumb-line can be dropped. A compass is sometimes attached to the table, or a detached compass, consisting of a needle in a narrow box, (called a Declinator), is placed upon it, as desired. The edges of the table are sometimes divided into degrees, like the "Drawing board Protractor," Art. (273). It then becomes a sort of Goniometer, like that of Art. (213).

(451) The Alidade. The ruler has a fiducial or feather edge, which may be divided into inches, tenths, &c. At each end it carries a sight like those of the compass. Two needles would be tolerable substitutes. The sights project beyond its edge so that their centre lines shall be precisely in the same vertical plane as this edge, in order that the lines drawn by it may correspond to the lines sighted on by them. To test this, fix a needle in the board, place the alidade against it, sight to some near point, draw a line by the ruler, turn it end for end, again place it against the needle, again sight to the same point, and draw a new line. If it coincides with the former line, the above condition is satisfied. The ruler and sights together take the name of Alidade. If a point should be too high or too low to be seen with the alidade, a plumb-line, held between the eye and the object, will remove the difficulty.

A telescope is sometimes substituted for the sights, being supported above the ruler by a standard, and capable of pointing upward or downward. It admits of adjustments similar in principle

to the 2d and 3d adjustments of the Transit, Part IV, Chapter 3, pages 242 and 246.

But even without these adjustments, whether of the sights or of the telescope, a survey could be made which would be perfectly correct as to the relative position of its parts, however far the line of sight might be from lying in the same vertical plane as the edge of the ruler, or even from being parallel to it; just as in the Transit or Theodolite the index or vernier need not to be exactly under the vertical hair of the telescope, since the angular deviation affects all the observed directions equally.

(452) Method of Radiation. This is the simplest, though not the best, method of surveying with the Plane-table. It is especially applicable to survey

Fig. 307.
B

head of sealing-wax) in the

board to represent the sta

tion. Direct the alidade to any corner of the field, as A, the fiducial edge of the ruler touching the needle, and draw an indefinite line by it. Measure OA, and set off the distance, to any desired scale, from the needle point, along the line just drawn, to a. The line OA is thus platted on the paper of the table as soon as determined in the field. Determine and plat in the same way, OB, OC, &c., to b, c, &c. Join ab, bc, &c., and a complete plat of the field is obtained. Trees, houses, hills, bends of rivers, &c., may be determnied in the same manner. The corresponding method with the Compass or Transit, was described in Articles (258) and (391). The table may be set at one of the angles of the field, if more convenient. If the alidade has a telescope, the method of measuring distances with a stadia, described in Art. (375), may be here applied with great advantage.

(453) Method of Progression. Let ABCD, &c., be the line

sents B, and which should be named b, shall be exactly over that station. Sight to A, pressing the fiducial edge of the ruler against the needle, and draw a line by it. Measure BA, and set off its length, to the desired scale, on the line just drawn, from 6 to a point a, representing A. Then sight to C, draw an indefinite line. by the ruler, and on it set off the length of BC from 6 to c. Fix the needle at c. Set up at C, the point c being over this station, and make the line cb of the plat coincide in direction with CB on the ground, by placing the edge of the ruler on cb, and turning the table till the sights point to B. The compass, if the table have one, will facilitate this. Then sight forward from C to D, and fix CD, cd on the plat, as be was fixed. Set up at D, make de coincide with DC, and proceed as before. The figure shews the lines drawn at each successive station. The Table drawn at A shews how the survey might be commenced there.

In going around a field, the work would be proved by the last line "closing" at the starting point, and, during the progress of the survey, by any direction, as from C to A on the ground, coinciding with the corresponding line, ca, on the plat.

This method is substantially the same as the method of surveying a line with the Transit, explained in Art. (372). It requires all the points to be accessible. It is especially suited to the survey of a road, a brook, a winding path through woods, &c. The offsets required may often be sketched in by eye with sufficient precision.