course, whether it be 0, or a meridian distance, both opposite that course and at the head of the longitude column.*

4. Add together in the Longitude column the head number, and the number opposite the first course, and place the sum in the Double Longitude column opposite the same course. Place the sum of the first and second numbers opposite the second course; the sum of the second and third opposite the third course, &c., through all the remaining sides.

5. Multiply the Double Longitude belonging to each course by its corresponding Northing or Southing; placing the product in the column of areas of the same name, and opposite that course. The difference between the sum of the North Areas, and the sum of the South Areas will be equal to twice the area of the Plot, in square units of the same name as the given distances. (Art. 54.) If the distances are chains, that difference will be twice the area of the field.

(57.) TEST OF WORK.

After the Longitudes, and the Northings and Southings of the several courses are obtained, they should be tested by remeasuring the former, and comparing the sums of the two latter, which should be equal. Indeed the entire operation of determining areas, like that of obtaining the field data, and plotting, requires great care. A mistake of one figure in the column of Double Longitudes, for instance, may occasion the owner of the field surveyed or some one else, the loss of hundreds of dollars, or a most perplexing and expensive lawsuit.

The use of the Trigonometer however, so greatly simplifies the methods of determining areas, and correcting errors, that with it the Surveyor need never allow an error to go out of his hands.

(58.) ADDITIONAL PLATE.

It is sometimes desirable in obtaining the Longitude, to place upon the Rule between the Attached Guide and the base

If the longitude is less than a unit, the first method of measuring (Art. 10), must be used.

of the Trigonometer a short metallic plate of exactly known length and breadth, in order either to throw the zero meridian farther from the edge of the table than it would otherwise fall, or to increase the available length of the Scale-Plate. But it should always be recollected that whenever used for the latter purpose, the distance employed, whether the length or breadth of the plate, must be added to the Longitude.

(59.) PLOTTING BY DOTS.

When the surveyor or student has made himself familiar with the method of determining areas by Latitude and Longitude, he will find it more expeditious to plot the field according to directions given (Arts. 25 and 42); viz.

First, plot the field by merely dotting with the prick the several angular points: enclosing in a little circle, with a common soft pencil, each dot when made, so that it may be readily seen, without drawing the sides at all, or even a meridian; the points with their corresponding intersecting parallels being designated by the numerals 1, 2, 3, 4, &c. The FIG. 26. FIG. 27.

Zero and Latitude meridians may, just as well, indeed better be supposed drawn than to be actually drawn.

These directions may be illustrated by the following example :

Ex. 4. Figs. 26, 27. Given the following courses and distances to find the area of the field. Fig. 26, gives the outline, Fig. 27, merely the angular points. The Zero and Latitude meridians in this example are identical.

(60.) In the following example the Zero Meridian is supposed drawn 1 chain east of the extreme eastern angular point of the field. The longitude of that point is therefore 1.25 chain; which by (Art. 56, 3), must be placed both opposite the last course, and at the head of the Longitude column. The courses and distances may be obtained from the data given in Fig. 20, (Art. 48.)

SHIFT OF BOARD.

(61.) If the length of a field is much greater than the breadth, and the sides corresponding with the former are within a few degrees of coincidence with a parallel of latitude passing through it, it is sometimes convenient in obtaining the data for the columns of Northings, Southings, and Longitude, to consider the sides of the Draughting Board shifted one fourth of a revolution to the right or left. The north or south sides then become the east or west sides.

The following example will illustrate the operation.

Let the board be supposed to revolve 90° from left to right. East then becomes North, West-South, South-East, and North-West; as in the subjoined table.

The advantage of this change is, that the longitude of the farther angular points being thus greatly diminished, that of each can be obtained by once setting the Vernier Guide, instead of twice.

SECTION IV.

LAND.

DIVIDING AND LAYING OUT LAND.

When it is required to lay out a given area in a rectangular form, the process being arithmetical, the Trigonometer is unnecessary, except in the following case.

PROBLEM I.

(62.) To lay out a given area in a rectangular form, having the length to exceed the breadth by a given difference.

With the instrument set at its meridian draw a straight line AB, of indefinite length; from an assumed point C in this line, lay off both ways half the given difference; from D the end of one of these, at right angles draw a line equal to the square root of the given area; join its latter terminus E, to the assumed point C. The half difference CB or CD added to CE, equals the length, and subtracted therefrom equals the breadth of the required parallelogram. With the data thus obtained the figure may be readily completed.

G

FIG. 28.

D

D

B

The following example illustrates the operation. Fig. 28.

It is required to lay out 48.8 acres in the form of a parallelogram, whose length is to exceed its breadth by 10 chains.

With the Trigonometer set at its meridian draw indefinitely the straight line AB; from an assumed point C in this line make CB and CD each equal to 5, half the given difference; from D, at Hright angles to CD draw DE,

equal to 22.09 chains, the square root of 488 square chains,