lines for the measurement of which are sketched out as below, are required.

D

B

Here the double area of the trapezium ABCD is found, as in Problem III, to which the double areas of the several offsets on the lines A B, BC, CD, DA must be added, and the sum will be the double area of the whole.

NOTE. A proof line, as AC should be measured, that the accuracy of the work may be insured :—but often pieces of ground are measured with the chain and cross that are never laid down, in this case great care should be taken in entering the field-notes, and in making the calculation.

5. Required the plan and area of a field from the following notes.

NOTE. The field in this case has also four sides, like the one proposed to te measured in the last example. Fields of this kind are often required to be measured in the rural districts, to ascertain the quantity of a growing crop, as grain, hay, turnips, &c., when sold by the acre; also to find the quantity of reaping, mowing, planting, &c.; in these cases the plan is seldom or never required, and the measurement is only taken as far as the growing crop extends, leaving out the hedges, ditches, and all other waste or other ground, not occupied by the crop in question.

6. Give the plan and area of a field from the following

TO FIND THE AREA OF A SEGMENT OF A CIRCLE, OR ANY OTHER CURVILINEAL FIGURE BY EQUIDISTANT OFFSETS OR ORDINATES.

a b c def g

A B C D E F G

RULE. If a right line AG be divided into any number of equal parts, AB, BC, CD, &c., and at the points of division perpendiculars be erected, Aa, Bb, Cc, &c., to the curve abcdefg; then to the sum of the first and last offsets, add four times the sum of all the even offsets, and twice the sum of all the odd offsets, not including the first and last; multiply the sum by the common distance of the offsets, and one-third of the product will be the area, recollecting that the second, fourth, &c., are the even offsets, and the third, fifth, &c., are the odd offsets.

NOTE. If any portion of the figure is not included by an even number of offsets, its area must be found separately and added to the area found by the Rule.

EXAMPLES.

1. Required the plan and area of a piece of land measured by equidistant offsets or ordinates, from the following dimensions. (See last figure.)

2. Required the plan and area of a piece of land, measured by equidistant offsets, the dimensions being as given below.

3. Plan and find the area of a piece of ground from the equidistant offsets given below.

4. Required the areas of two fields, the ends of which are straight and parallel, and the side curved by the following equidistant offsets.

TO MEASURE A LINE ACROSS A WIDE RIVER.

Let the annexed figure be a river, which is required to be crossed by the chain-line PB. Fix, or cause to be fixed, a pole or mark at B, at or near the margin of the river, in the line to be measured; erect the perpendicular A D, measuring A C and CD of any equal lengths; at D erect the perpendicular DE; on arriving at E, in the direction of B C produced, the distance D E will be equal to A B, the required breadth of the river.

A

Pi

B

From the arrangement of the lines in the figure, it is evident that the triangles CA B, C D E are equiangular, and since A C was made CD, the triangles are equal in all respects, and consequently A B = DE.

=

NOTE. We have thus given in sufficient detail the mode of surveying with the cross, which, though not much used by experienced surveyors, is a simple instrument, and its use readily understood by students. This method is, therefore, a proper introduction to the higher branches of surveying; besides, in rural districts, villages, &c., few surveyors use the more expensive instrument, the chain and cross being found quite sufficient to measure the quantities of growing crops, and other such small surveys as may be there required.