NOTE. To draw a tree, house, tower, or any other remarkable object, in its proper place, in the plot of a field-from any two stations, while surveying the field, take the bearing of an object, and the intersection of the lines, which represent the bearings, will determine the place of the object, in the same manner that the tower is

TO FIND THE AREA OF THE ABOVE FIELD.

Find the area within the stationary lines, and then of the several small trapezoids, &c. remembering to distinguish those without the stationary lines from those which are within. Substract the area of those within the stationary lines from the area of those without, and add the remainder to the area contained within the stationary lines; the sum will be the whole area of the field.

[Or, add the areas of those without the stationary lines, to the area contained within those lines, and substract from the sum the areas of the several triangles, trapezoids, &c. within the stationary lines.]

SECTION III.

RECTANGULAR SURVEYING, OR AN ACCURATE METHOD of CALCULATING THE AREA OF A FIELD ARITHMETICALLY, FROM THE FIELD BOOK, WITHOUT THE NECESSITY OF PROTRACTING IT, AND MEASURING WITH A SCALE AND DIVIDERS, AS IS COMMONLY PRACTISED.

I. Survey the field in the usual method, with an accurate compass and chain, and from the field book set down, in a traverse table, the course or bearing of the several sides, and their length in chains and links, or rods and decimal parts of a rod; as in the 2nd and 3d columns of the following Ex

AMPLE.

2. Calculate by RIGHT ANGLED TRIGONOMETRY, CASE I, or find by the table of difference of latitude and departure,* or by the table of natural sines,† the northing or southing, easting or westing, made on each course, and set them down against their several courses in their proper columns, marked N. S. E. W.

NOTE. To determine whether the latitude and departure for any particular course and distance are accurately calculated, square each of them; and if they are right, the sum of their squares will equal the square of the distance, for the following reason: the latitude and departure represent the two legs of a right angled triangle, and the distance the hypothenuse; and it is a mathematical truth, that the square of the hypothenuse of any right angled triangle, is equal to the sum of the squares of the two legs.

3. If the survey has been accurately taken, the sum of the northings will equal that of the southings, and the sum of the eastings will equal that of the westings. If, upon adding up the respective columns, these are found to differ very considerably, the field should be again surveyed; as some error must have been committed, either in taking the courses, or measuring the sides. If the difference is small, a judicious, experienced surveyor, will judge from the nature of the ground, or shape of the field surveyed, where the mistake was most probably made, and will correct accordingly. Or, the northings and southings, and the eastings and westings, may be equalled by balancing them, as follows: substract one half the difference from that column which is the largest, and add it to that column which is the smallest; and let the difference, to be added or substracted, be divided among the several courses, according to their length.

In EXAMPLE I., the upper numbers are the northings, &c., as found by a table of difference of latitude and departure. The several columns being added, the northings are found to exceed the southings 47 links, and the westings to exceed the eastings 24 links; [47 being uneven, drop a link from the northings, and it becomes 46. Let half of this (23) be taken from the northings, and added to the southings;] likewise, take 12 links from the westings, and add it to the eastings. Take from the first course of the northings 12 links, from the second 7, and from the third 5; to the first southing add 7 links, to the second 10, and to the third 6; add to the first easting 3 links, to the second 3, to the third 4, and to the

For an explanation of this table, and the manner of using it, see the remarks preceding the table.

fourth 2; take from the first westing 5 links, from the second 4, and from the third 3. [These are the proportional corrections belonging to each, as found by calculation.*] The lower numbers will then represent the northings, &c. as balanced.

4. These columns being balanced, proceed to form a departure column, or a column of meridian distances; which shows how far the end of each side of the field is east or west of the station where the calculation begins. This column is formed by a continual addition of the eastings, and substraction of the westings; or by adding the westings and substracting the eastings: see EXAMPLE 1.

The first easting, 20.74, is set for the first number in the departure column; to this add 24.38, the second easting, and it makes 45.12, for the second number; to this add 30.04, the third easting, and it makes 75.16, for the third number; to this add 9.56, the fourth easting, and it makes 84.72, for the fourth number; the fifth course being south, it is evident the meridian distance will remain the same, therefore place against it the same easting as for the prece ding course; from this substract 39.95, the first westing, and it leaves 44.77, for the sixth course; from this substract 23.75, the second westing, and it leaves 21.02, for the seventh course; from this substract 21.02, the last westing, and it leaves 0.0 to be set against the last course, which shows that the additions and substractions have been accurately made. For as the eastings and westings equal each other, it is evident that the one being added and the other substracted, there will be in the end no remainder.

5. The next step in the process is to form a second departure column, the numbers in which show the sum of the meridian distances at the end of the first and second, second and third, third and fourth courses, &c.

The first number in this column will be the first in the other departure column; to which add the second number in that column for the second in this; for the third add the second and third; and for the fourth the third and fourth; and so on, until the column be completed. See EXAMPLE I.

The first number to be placed in the second departure column is 20.74; to this add 45.12, and it makes 65.86 for the second number; to 45.12 add 75.16, and it makes 120.28, for the third number; to 75.16 add 84.72, and it makes 159.88 for the fourth number; to 84.72 add 84.72, and it makes 169.44 for the fifth number; to 84.72, add 44.77, and it makes