Example 2.-The length of an irregular figure being 84, and the breadths, at six equidistant places, 17.4, 20.6, 14.2, 16.5, 20.1, 24.4 rods; required the area. Ans.-9 acres, 2 roods, 30.64 perches.

PROBLEM.

116. To find the area of a circle.

Square the radius and multiply the number so obtained by 3.1416, the product is the required area. (291.*)

PROBLEM.

117. To find the area of an ellipsis.

Multiply the axes together, and their product by .7854, the last product will be the area of the ellipsis.

Example. Required the area of an ellipsis, whose two axes are 70 and 50 poles. Ans.--17 acres, 28.9 perches.

118. With respect to the content of land, it may be remarked, that irregular pieces must, in general, be divided into triangles, rectangles, parallelograms, or trapezoids. The manner of making such divisions, depends on the lines and angles which are known. They should always, however, be so made as to simplify the calculations, and so that a sufficient number of known parts may fall into each figure, to

CHAPTER IV.

OF THE INSTRUMENTS USED IN SURVEYING.

119. THE measurements which are necessary on the field, and which furnish the data for the computation of the area of ground, are generally made with the theodolite and chain, the surveying cross, the plain table, and the compass. The uses of the theodolite and chain have already been explained, the remaining instruments are now to be described.

We shall also describe, in this chapter, the drawing instruments used for plotting a survey: they are, besides the dividers and common rule, the semicircular protractor, the circular protractor, the scale of chords, the diagonal scale of equal parts, the sectoral scale of equal parts, and Gunter's scale.

OF THE SURVEYING CROSS.

120. This instrument consists of two bars, AB and CD, (Pl. 4, Fig. 1,) permanently fixed at right angles to each other, and firmly attached at E to a pointed staff, which serves as a support. Four sights are screwed firmly to the bars by means of the screws a, b, c, and d. As the only use of this instrument is to lay off right angles, it is of the first importance that the lines of sight be truly at right angles. This can easily be ascertained: for, let the bar AB be turned until its sights mark some distinct object; then look through the other sights and place a staff on the line which they indicate: let the cross then be turned until the sights of the bar AB come to the same line: if the other sights are directed to the first object, the lines of sight are exactly at right angles.

The sights being at right angles, one of them being turned

tion of a line perpendicular to it, at the point where the instrument is placed.

OF THE PLAIN TABLE.

121. This instrument consists of two parts, a rectangular board CDBA, (Pl. 3, Fig. 1,) and a tripod EHG, to which it is firmly screwed.

Directly under the rectangular board are four milled screws that pass through sockets inserted in a horizontal brass plate, and which are worked against a second horizontal plate, for the purpose of levelling the table: the table having a ball and socket motion, similar to the limb of the theodolite.

For the purpose of levelling the table, a small detached spirit level is used. This level being placed over the centre, and also over two of the levelling screws, the screws are turned contrariwise until the level is horizontal; after which, it is placed over the other two screws, and made horizontal in the same manner.

Between the upper horizontal plate and the table there is a clamp screw, similar to the clamp screw of the theodolite, which being loosened, the table can be revolved freely abou: its axis. There is, also, a small tangent screw, by which the smaller motions of the table are regulated after the clamp screw is made fast. Neither of these screws can be seen in

the figure.

The upper side of the table is bordered by four brass plates, about one inch in width, and the centre of the table is marked by a small pin, F. If around this centre a circumscribing circle be described, its circumference, divided into degrees and parts of a degree, and radii drawn through the points of division, the points where such radii would intersect the outer edge of the brass border are marked by lines on the brass plates, and the degrees are numbered in the direction from left to right, from the point L to the point I, 180°, and from the point I to the point L, 180°. In some Plain Tables, however, they are numbered from 0 to 360°.

There are, generally, diagonal scales of equal parts cut on

explained hereafter. Near the other two 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, 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 also sometimes used with the plain table.

A magnetic needle and compass are sometimes attached to the plain table, either by screws, or by fixing it directly under its centre, to show the direction 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.

122. To measure horizontal angles.-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 it 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 0

if it has, the larger taken from 180°, and the remainder added to the smaller gives the required angle.

123. Of the determination of lines in extent and position.— Having placed a paper on the table, examine the objects and lines, whose positions and lengths 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. Place the fiducial edge of the ruler AB, against the pin, it being held firmly, sight to the other extremity of the base line, and mark with the pin or pencil the direction of the ruler 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, 4, &c.

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 then be levelled, after which, place the ruler again on the base line, and bring the table to its proper position by the tangent screw, and so 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, and the table truly