Line HI, and the Product 25168 Square Links will be the Area of the Trapezoid HI32. Proceed in the same manner to calculate the Area of all the Trapezoids, Triangles, &c.

CASE X.

To survey a Field by taking Offsets both to the Right and Left; that is, within and without the Field, as occasion shall require, in consequence of the stationary Lines crossing the boundary Lines: Also, by Intersections, that is, taking the Bearing of an inaccessible Corner from two Stations.

The directions given in the preceding CASE, together with the following FIELD BOOK, will show the Learner how to survey a Field like the following, and also to protract it when surveyed.

FIELD BOOK. See PLATE IV. Fig. 69.

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 the 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 drawn in the Figure.

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. Subtract 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.

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.

1. 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 2d and 3d Columns of the following EXAMPLE.

2. Calculate by RIGHT NGLED TRIGONOMETRY,

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 Hy-pothenuse; 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 the Southings; and the Eastings will equal 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: Subtract one half the difference from that Column which is the largest, and add the other half to that Column which is the smallest; and let the difference to be added or subtracted be divided among the several Courses according to their length.

*For an Explanation of this Table, and the manner of using it, see the Remarks preceding the Table.

+ See the Remarks preceding the Table of Natural Sines.

A method of determining whether the Courses are right has been already explained. See page 50. The Surveyor, before he leaves the Field, should calculate the Northings, Southings, &c. and if he finds much difference determine whether the Courses are right. This will show him whether a re-survey is necessary, and will enable him to ascertain whether the error lies in the

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. They may be balanced by taking 24 Links from the Northings, and adding 23 Links to the Southings; and taking 12 Links from the Westings and adding 12 Links 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 fourth 2; take from the first Westing 5 Links, from the second 4, and from the third 3. 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 Eastor West of the Station where the calculation begins. This Column is formed by a continual addition of the Eastings and subtraction of the Westings; or by adding the Westings and subtracting the Eastings. See EXAMPLE I.

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 preceding Course; from this subtract 39.95 the first Westing, and it leaves 44.77 for the sixth Course; from this subtract 23.75 the second Westing, and it leaves 21.02 for the seventh Course; from this subtract 21.02 the last Westing, and it leaves 0.0 to be set against the last course, which shows that the additions and subtractions have been ac